math problem solving inequalities

Solving Inequalities

Sometimes we need to solve Inequalities like these:

Our aim is to have x (or whatever the variable is) on its own on the left of the inequality sign:

We call that "solved".

Example: x + 2 > 12

Subtract 2 from both sides:

x + 2 − 2 > 12 − 2

x > 10

How to Solve

Solving inequalities is very like solving equations , we do most of the same things ...

... but we must also pay attention to the direction of the inequality .

Some things can change the direction !

< becomes >

> becomes <

≤ becomes ≥

≥ becomes ≤

Safe Things To Do

These things do not affect the direction of the inequality:

• Add (or subtract) a number from both sides
• Multiply (or divide) both sides by a positive number
• Simplify a side

Example: 3x < 7+3

We can simplify 7+3 without affecting the inequality:

But these things do change the direction of the inequality ("<" becomes ">" for example):

• Multiply (or divide) both sides by a negative number
• Swapping left and right hand sides

Example: 2y+7 < 12

When we swap the left and right hand sides, we must also change the direction of the inequality :

12 > 2y+7

Here are the details:

Adding or Subtracting a Value

We can often solve inequalities by adding (or subtracting) a number from both sides (just as in Introduction to Algebra ), like this:

Example: x + 3 < 7

If we subtract 3 from both sides, we get:

x + 3 − 3 < 7 − 3    

And that is our solution: x < 4

In other words, x can be any value less than 4.

What did we do?

And that works well for adding and subtracting , because if we add (or subtract) the same amount from both sides, it does not affect the inequality

Example: Alex has more coins than Billy. If both Alex and Billy get three more coins each, Alex will still have more coins than Billy.

What If I Solve It, But "x" Is On The Right?

No matter, just swap sides, but reverse the sign so it still "points at" the correct value!

Example: 12 < x + 5

If we subtract 5 from both sides, we get:

12 − 5 < x + 5 − 5    

That is a solution!

But it is normal to put "x" on the left hand side ...

... so let us flip sides (and the inequality sign!):

Do you see how the inequality sign still "points at" the smaller value (7) ?

And that is our solution: x > 7

Note: "x" can be on the right, but people usually like to see it on the left hand side.

Multiplying or Dividing by a Value

Another thing we do is multiply or divide both sides by a value (just as in Algebra - Multiplying ).

But we need to be a bit more careful (as you will see).

Positive Values

Everything is fine if we want to multiply or divide by a positive number :

Example: 3y < 15

If we divide both sides by 3 we get:

3y /3 < 15 /3

And that is our solution: y < 5

Negative Values

Well, just look at the number line!

For example, from 3 to 7 is an increase , but from −3 to −7 is a decrease.

See how the inequality sign reverses (from < to >) ?

Let us try an example:

Example: −2y < −8

Let us divide both sides by −2 ... and reverse the inequality !

−2y < −8

−2y /−2 > −8 /−2

And that is the correct solution: y > 4

(Note that I reversed the inequality on the same line I divided by the negative number.)

So, just remember:

When multiplying or dividing by a negative number, reverse the inequality

Multiplying or Dividing by Variables

Here is another (tricky!) example:

Example: bx < 3b

It seems easy just to divide both sides by b , which gives us:

... but wait ... if b is negative we need to reverse the inequality like this:

But we don't know if b is positive or negative, so we can't answer this one !

To help you understand, imagine replacing b with 1 or −1 in the example of bx < 3b :

• if b is 1 , then the answer is x < 3
• but if b is −1 , then we are solving −x < −3 , and the answer is x > 3

The answer could be x < 3 or x > 3 and we can't choose because we don't know b .

Do not try dividing by a variable to solve an inequality (unless you know the variable is always positive, or always negative).

A Bigger Example

Example: x−3 2 < −5.

First, let us clear out the "/2" by multiplying both sides by 2.

Because we are multiplying by a positive number, the inequalities will not change.

x−3 2 ×2 < −5  ×2  

x−3 < −10

Now add 3 to both sides:

x−3 + 3 < −10 + 3    

And that is our solution: x < −7

Two Inequalities At Once!

How do we solve something with two inequalities at once?

Example: −2 < 6−2x 3 < 4

First, let us clear out the "/3" by multiplying each part by 3.

Because we are multiplying by a positive number, the inequalities don't change:

−6 < 6−2x < 12

−12 < −2x < 6

Now divide each part by 2 (a positive number, so again the inequalities don't change):

−6 < −x < 3

Now multiply each part by −1. Because we are multiplying by a negative number, the inequalities change direction .

6 > x > −3

And that is the solution!

But to be neat it is better to have the smaller number on the left, larger on the right. So let us swap them over (and make sure the inequalities point correctly):

−3 < x < 6

• Many simple inequalities can be solved by adding, subtracting, multiplying or dividing both sides until you are left with the variable on its own.
• Multiplying or dividing both sides by a negative number
• Don't multiply or divide by a variable (unless you know it is always positive or always negative)

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1.5: Solve Inequalities

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Learning Objectives

Represent inequalities on a number line, represent inequalities using interval notation.

• Use the addition and multiplication properties to solve algebraic inequalities and express their solutions graphically and with interval notation
• Solve inequalities that contain absolute value
• Combine properties of inequality to isolate variables, solve algebraic inequalities, and express their solutions graphically
• Simplify and solve algebraic inequalities using the distributive property to clear parentheses and fractions

First, let’s define some important terminology. An inequality is a mathematical statement that compares two expressions using the ideas of greater than or less than. Special symbols are used in these statements. When you read an inequality, read it from left to right—just like reading text on a page. In algebra, inequalities are used to describe large sets of solutions. Sometimes there are an infinite amount of numbers that will satisfy an inequality, so rather than try to list off an infinite amount of numbers, we have developed some ways to describe very large lists in succinct ways.

The first way you are probably familiar with—the basic inequality. For example:

• \({x}\lt{9}\) or try to list all the possible numbers that are less than 9? (hopefully, your answer is no)

Note how placing the variable on the left or right of the inequality sign can change whether you are looking for greater than or less than.

For example:

• \(x\lt5\) means all the real numbers that are less than 5, whereas;
• \(x\gt{5}\) note how the inequality is still pointing the same direction relative to x. This statement represents all the real numbers that are greater than 5, which is easier to interpret than 5 is less than x.

The second way is with a graph using the number line:

And the third way is with an interval.

We will explore the second and third ways in depth in this section. Again, those three ways to write solutions to inequalities are:

• an inequality
• an interval

Inequality Signs

The box below shows the symbol, meaning, and an example for each inequality sign. Sometimes it’s easy to get tangled up in inequalities, just remember to read them from left to right.

The inequality \({y}<{x}\). The sides of any inequality can be switched as long as the inequality symbol between them is also reversed.

Graphing an Inequality

Inequalities can also be graphed on a number line. Below are three examples of inequalities and their graphs. Graphs are a very helpful way to visualize information – especially when that information represents an infinite list of numbers!

\(x\leq -4\). This translates to all the real numbers on a number line that are less than or equal to 4.

\({x}\geq{-3}\). This translates to all the real numbers on the number line that are greater than or equal to -3.

Each of these graphs begins with a circle—either an open or closed (shaded) circle. This point is often called the end point of the solution. A closed, or shaded, circle is used to represent the inequalities greater than or equal to \(\displaystyle \left(\leq\right)\). The point is part of the solution. An open circle is used for greater than (>) or less than (<). The point is not part of the solution.

The graph then extends endlessly in one direction. This is shown by a line with an arrow at the end. For example, notice that for the graph of \(−3\), represented with a closed circle since the inequality is greater than or equal to \(−3\). The arrow at the end indicates that the solutions continue infinitely.

Graph the inequality \(x\ge 4\) [reveal-answer q=”797241″]Show Solution[/reveal-answer] [hidden-answer a=”797241″]

We can use a number line as shown. Because the values for x include 4, we place a solid dot on the number line at 4.

[/hidden-answer]

This video shows an example of how to draw the graph of an inequality.

A YouTube element has been excluded from this version of the text. You can view it online here: pb.libretexts.org/ba/?p=52

Write an inequality describing all the real numbers on the number line that are less than 2, then draw the corresponding graph. [reveal-answer q=”867890″]Show Solution[/reveal-answer] [hidden-answer a=”867890″]

We need to start from the left and work right, so we start from negative infinity and end at \(-2\).

Inequality: \(x<2\)

To draw the graph, place an open dot on the number line first, then draw a line extending to the left. Draw an arrow at the leftmost point of the line to indicate that it continues for infinity.

The following video shows how to write an inequality mathematically when it is given in words. We will then graph it.

Another commonly used, and arguably the most concise, method for describing inequalities and solutions to inequalities is called interval notation. With this convention, sets are built with parentheses or brackets, each having a distinct meaning. The solutions to \(\left[4,\infty \right)\). This method is widely used and will be present in other math courses you may take.

The main concept to remember is that parentheses represent solutions greater or less than the number, and brackets represent solutions that are greater than or equal to or less than or equal to the number. Use parentheses to represent infinity or negative infinity, since positive and negative infinity are not numbers in the usual sense of the word and, therefore, cannot be “equaled.” A few examples of an interval , or a set of numbers in which a solution falls, are \(-2\) and \(-2\), but not including \(\left(-1,0\right)\), all real numbers between, but not including \(0\); and \(1\). The table below outlines the possibilities. Remember to read inequalities from left to right, just like text.

The table below describes all the possible inequalities that can occur and how to write them using interval notation, where a and b are real numbers.

Describe the inequality \(x\ge 4\) using interval notation [reveal-answer q=”817362″]Show Solution[/reveal-answer] [hidden-answer a=”817362″]

The solutions to \(\left[4,\infty \right)\).

Note the use of a bracket on the left because 4 is included in the solution set.

In the following video we show another example of using interval notation to describe an inequality.

Use interval notation to indicate all real numbers greater than or equal to \(-2\). [reveal-answer q=”961990″]Show Solution[/reveal-answer] [hidden-answer a=”961990″]

Use a bracket on the left of \(\left[-2,\infty \right)\). The bracket indicates that \(-2\) to infinity.

In the following video we show another example of translating words into an inequality and writing it in interval notation, as well as drawing the graph.

Think About It

In the previous examples you were given an inequality or a description of one with words and asked to draw the corresponding graph and write the interval. In this example you are given an interval and asked to write the inequality and draw the graph.

Given \(\left(-\infty,10\right)\), write the associated inequality and draw the graph.

In the box below, write down whether you think it will be easier to draw the graph first or write the inequality first.

[practice-area rows=”1″][/practice-area] [reveal-answer q=”15120″]Show Solution[/reveal-answer]

[hidden-answer a=”15120″]

We will draw the graph first.

The interval reads “all real numbers less than 10,” so we will start by placing an open dot on 10 and drawing a line to the left with an arrow indicating the solution continues to negative infinity.

To write the inequality, we will use < since the parentheses indicate that 10 is not included. \(x<10\)

In the following video, you will see examples of how to draw a graph given an inequality in interval notation.

And finally, one last video that shows how to write inequalities using a graph, with interval notation and as an inequality.

Solve Single-Step Inequalities

Solve inequalities with addition and subtraction.

You can solve most inequalities using inverse operations as you did for solving equations. This is because when you add or subtract the same value from both sides of an inequality, you have maintained the inequality. These properties are outlined in the box below.

Addition and Subtraction Properties of Inequality

If \(a+c>b+c\).

If \(a−c>b−c\).

Because inequalities have multiple possible solutions, representing the solutions graphically provides a helpful visual of the situation, as we saw in the last section. The example below shows the steps to solve and graph an inequality and express the solution using interval notation.

Solve for x.

\({x}+3\lt{5}\)

[reveal-answer q=”952771″]Show Solution[/reveal-answer] [hidden-answer a=”952771″]

It is helpful to think of this inequality as asking you to find all the values for x , including negative numbers, such that when you add three you will get a number less than 5.

\(\displaystyle \begin{array}{l}x+3<\,\,\,\,5\\\underline{\,\,\,\,\,-3\,\,\,\,-3}\\x\,\,\,\,\,\,\,\,<\,\,\,\,2\,\,\end{array}\)

Isolate the variable by subtracting 3 from both sides of the inequality.

Interval: \(\left(-\infty, 2\right)\)

The line represents all the numbers to which you can add 3 and get a number that is less than 5. There’s a lot of numbers that solve this inequality!

Just as you can check the solution to an equation, you can check a solution to an inequality. First, you check the end point by substituting it in the related equation. Then you check to see if the inequality is correct by substituting any other solution to see if it is one of the solutions. Because there are multiple solutions, it is a good practice to check more than one of the possible solutions. This can also help you check that your graph is correct.

The example below shows how you could check that \(x+3<5\) .

Check that \(x+3<5\).

[reveal-answer q=”811564″]Show Solution[/reveal-answer] [hidden-answer a=”811564″]

Substitute the end point 2 into the related equation, \(x+3=5\).

\(\begin{array}{r}x+3=5 \\ 2+3=5 \\ 5=5\end{array}\)

Pick a value less than 2, such as 0, to check into the inequality. (This value will be on the shaded part of the graph.)

\(\displaystyle \begin{array}{r}x+3<5 \\ 0+3<5 \\ 3<5\end{array}\)

\(x+3<5\).[/hidden-answer]

The following examples show inequality problems that include operations with negative numbers. The graph of the solution to the inequality is also shown. Remember to check the solution. This is a good habit to build!

Solve for x : \(x-10\leq-12\) [reveal-answer q=”815894″]Show Solution[/reveal-answer] [hidden-answer a=”815894″]

Isolate the variable by adding 10 to both sides of the inequality.

\(\displaystyle \begin{array}{r}x-10\le -12\\\underline{\,\,\,+10\,\,\,\,\,+10}\\x\,\,\,\,\,\,\,\,\,\,\le \,\,\,-2\end{array}\)

Check the solution to \(x-10\leq -12\) [reveal-answer q=”268062″]Show Solution[/reveal-answer] [hidden-answer a=”268062″]

Substitute the end point \(x-10=−12\)

\(\displaystyle \begin{array}{r}x-10=-12\,\,\,\\\text{Does}\,\,\,-2-10=-12?\\-12=-12\,\,\,\end{array}\)

Pick a value less than \(−5\), to check in the inequality. (This value will be on the shaded part of the graph.)

\(\displaystyle \begin{array}{r}x-10\le -12\,\,\,\\\text{ }\,\text{ Is}\,\,-5-10\le -12?\\-15\le -12\,\,\,\\\text{It}\,\text{checks!}\end{array}\)

\(x-10\leq -12\)

Solve for a . \(a-17>-17\) [reveal-answer q=”343031″]Show Solution[/reveal-answer] [hidden-answer a=”343031″]

Isolate the variable by adding 17 to both sides of the inequality.

\(\displaystyle \begin{array}{r}a-17>-17\\\underline{\,\,\,+17\,\,\,\,\,+17}\\a\,\,\,\,\,\,\,\,\,\,\,>\,\,\,\,\,\,0\end{array}\)

Inequality: \(\displaystyle a\,\,>\,0\)

Interval: \(\left(0,\infty\right)\) Note how we use parentheses on the left to show that the solution does not include 0.

Graph: Note the open circle to show that the solution does not include 0.

Check the solution to \(a-17>-17\) [reveal-answer q=”653357″]Show Solution[/reveal-answer] [hidden-answer a=”653357″]

Is \(a-17>-17\)?

Substitute the end point 0 into the related equation.

\(\displaystyle \begin{array}{r}a-17=-17\,\,\,\\\text{Does}\,\,\,0-17=-17?\\-17=-17\,\,\,\end{array}\)

Pick a value greater than 0, such as 20, to check in the inequality. (This value will be on the shaded part of the graph.)

\(\displaystyle \begin{array}{r}a-17>-17\,\,\,\\\text{Is }\,\,20-17>-17?\\3>-17\,\,\,\\\\\text{It checks!}\,\,\,\,\end{array}\)

\(a-17>-17\)

The previous examples showed you how to solve a one-step inequality with the variable on the left hand side. The following video provides examples of how to solve the same type of inequality.

What would you do if the variable were on the right side of the inequality? In the following example, you will see how to handle this scenario.

Solve for x : \(4\geq{x}+5\) [reveal-answer q=”815893″]Show Solution[/reveal-answer] [hidden-answer a=”815893″]

\(\displaystyle \begin{array}{r}4\geq{x}+5 \\\underline{\,\,\,-5\,\,\,\,\,-5}\\-1\,\,\,\,\,\,\,\,\,\,\ge \,\,\,x\end{array}\)

Rewrite the inequality with the variable on the left – this makes writing the interval and drawing the graph easier.

\(x\le{-1}\)

Note how the the pointy part of the inequality is still directed at the variable, so instead of reading as negative one is greater or equal to x, it now reads as x is less than or equal to negative one.

Check the solution to \(4\geq{x}+5\) [reveal-answer q=”568062″]Show Solution[/reveal-answer] [hidden-answer a=”568062″]

Substitute the end point \(4=x+5\)

\(\displaystyle \begin{array}{r}4=x+5\,\,\,\\\text{Does}\,\,\,4=-1+5?\\-1=-1\,\,\,\end{array}\)

\(\displaystyle \begin{array}{r}4\geq{-5}+5\,\,\,\\\text{ }\,\text{ Is}\,\,4\ge 0?\\\text{It}\,\text{checks!}\end{array}\)

\(4\geq{x}+5\) [/hidden-answer]

The following video show examples of solving inequalities with the variable on the right side.

Solve inequalities with multiplication and division

Solving an inequality with a variable that has a coefficient other than 1 usually involves multiplication or division. The steps are like solving one-step equations involving multiplication or division EXCEPT for the inequality sign. Let’s look at what happens to the inequality when you multiply or divide each side by the same number.

Multiplication and Division Properties of Inequality

Keep in mind that you only change the sign when you are multiplying and dividing by a negative number. If you add or subtract by a negative number, the inequality stays the same.

Solve for x. \(3x>12\)

[reveal-answer q=”691711″]Show Solution[/reveal-answer] [hidden-answer a=”691711″]Divide both sides by 3 to isolate the variable.

\(\displaystyle \begin{array}{r}\underline{3x}>\underline{12}\\3\,\,\,\,\,\,\,\,\,\,\,\,3\\x>4\,\,\,\end{array}\)

Check your solution by first checking the end point 4, and then checking another solution for the inequality.

\(\begin{array}{r}3\cdot4=12\\12=12\\3\cdot10>12\\30>12\\\text{It checks!}\end{array}\)

Inequality: \(\displaystyle x>4\)

Interval: \(\left(4,\infty\right)\)

There was no need to make any changes to the inequality sign because both sides of the inequality were divided by positive 3. In the next example, there is division by a negative number, so there is an additional step in the solution!

Solve for x . \(−2x>6\)

[reveal-answer q=”604033″]Show Solution[/reveal-answer] [hidden-answer a=”604033″]Divide each side of the inequality by \(−2\) to isolate the variable, and change the direction of the inequality sign because of the division by a negative number.

\(\displaystyle \begin{array}{r}\underline{-2x}<\underline{\,6\,}\\-2\,\,\,\,-2\,\\x<-3\end{array}\)

Check your solution by first checking the end point \(−3\), and then checking another solution for the inequality.

\(\begin{array}{r}-2\left(-3\right)=6 \\6=6\\ -2\left(-6\right)>6 \\ 12>6\end{array}\)

Inequality: \(\displaystyle x<-3\)

Interval: \(\left(-\infty, -3\right)\)

Graph: \(−2\), the inequality symbol was switched from > to <. [/hidden-answer]

The following video shows examples of solving one step inequalities using the multiplication property of equality where the variable is on the left hand side.

Before you read the solution to the next example, think about what properties of inequalities you may need to use to solve the inequality. What is different about this example from the previous one? Write your ideas in the box below.

Solve for x . \(-\frac{1}{2}>-12x\)

[practice-area rows=”1″][/practice-area] [reveal-answer q=”811465″]Show Solution[/reveal-answer] [hidden-answer a=”811465″]

This inequality has the variable on the right hand side, which is different from the previous examples. Start the solution process as before, and at the end, you can move the variable to the left to write the final solution.

Divide both sides by \(-12\) to isolate the variable. Since you are dividing by a negative number, you need to change the direction of the inequality sign.

\(\displaystyle\begin{array}{l}-\frac{1}{2}\gt{-12x}\\\\\frac{-\frac{1}{2}}{-12}\gt\frac{-12x}{-12}\\\end{array}\)

Dividing a fraction by an integer requires you to multiply by the reciprocal, and the reciprocal of \(\frac{1}{-12}\)

\(\displaystyle\begin{array}{r}\left(-\frac{1}{12}\right)\left(-\frac{1}{2}\right)\lt\frac{-12x}{-12}\,\,\\\\ \frac{1}{24}\lt\frac{\cancel{-12}x}{\cancel{-12}}\\\\ \frac{1}{24}\lt{x}\,\,\,\,\,\,\,\,\,\,\end{array}\)

Inequality: \(x\gt\frac{1}{24}\). Writing the inequality with the variable on the left requires a little thinking, but helps you write the interval and draw the graph correctly.

Interval: \(\left(\frac{1}{24},\infty\right)\)

The following video gives examples of how to solve an inequality with the multiplication property of equality where the variable is on the right hand side.

Combine properties of inequality to solve algebraic inequalities

A popular strategy for solving equations, isolating the variable, also applies to solving inequalities. By adding, subtracting, multiplying and/or dividing, you can rewrite the inequality so that the variable is on one side and everything else is on the other. As with one-step inequalities, the solutions to multi-step inequalities can be graphed on a number line.

Solve for p . \(4p+5<29\)

[reveal-answer q=”211828″]Show Solution[/reveal-answer] [hidden-answer a=”211828″]

Begin to isolate the variable by subtracting 5 from both sides of the inequality.

\(\displaystyle \begin{array}{l}4p+5<\,\,\,29\\\underline{\,\,\,\,\,\,\,\,\,-5\,\,\,\,\,-5}\\4p\,\,\,\,\,\,\,\,\,<\,\,24\,\,\end{array}\)

Divide both sides of the inequality by 4 to express the variable with a coefficient of 1.

\(\begin{array}{l}\underline{4p}\,<\,\,\underline{24}\,\,\\\,4\,\,\,\,<\,\,4\\\,\,\,\,\,p<6\end{array}\)

Inequality: \(p<6\)

Interval: \(\left(-\infty,6\right)\)

Graph: Note the open circle at the end point 6 to show that solutions to the inequality do not include 6. The values where p is less than 6 are found all along the number line to the left of 6.

Check the solution. [reveal-answer q=”291597″]Show Solution[/reveal-answer] [hidden-answer a=”291597″]

Check the end point 6 in the related equation.

\(\displaystyle \begin{array}{r}4p+5=29\,\,\,\\\text{Does}\,\,\,4(6)+5=29?\\24+5=29\,\,\,\\29=29\,\,\,\\\text{Yes!}\,\,\,\,\,\,\end{array}\)

Try another value to check the inequality. Let’s use \(p=0\).

\(\displaystyle \begin{array}{r}4p+5<29\,\,\,\\\text{Is}\,\,\,4(0)+5<29?\\0+5<29\,\,\,\\5<29\,\,\,\\\text{Yes!}\,\,\,\,\,\end{array}\)

\(4p+5<29\)

Solve for x : \(3x–7\ge 41\) [reveal-answer q=”238157″]Show Solution[/reveal-answer] [hidden-answer a=”238157″]

Begin to isolate the variable by adding 7 to both sides of the inequality, then divide both sides of the inequality by 3 to express the variable with a coefficient of 1.

\(\displaystyle \begin{array}{l}3x-7\ge 41\\\underline{\,\,\,\,\,\,\,+7\,\,\,\,+7}\\\frac{3x}{3}\,\,\,\,\,\,\,\,\ge \frac{48}{3}\\\,\,\,\,\,\,\,\,\,\,x\ge 16\end{array}\)

Inequality: \(x\ge 16\)

Interval: \(\left[16,\infty\right)\)

Check the solution. [reveal-answer q=”437341″]Show Solution[/reveal-answer]

[hidden-answer a=”437341″]

First, check the end point 16 in the related equation.

\(\displaystyle \begin{array}{r}3x-7=41\,\,\,\\\text{Does}\,\,\,3(16)-7=41?\\48-7=41\,\,\,\\41=41\,\,\,\\\text{Yes!}\,\,\,\,\,\end{array}\)

Then, try another value to check the inequality. Let’s use \(x = 20\).

\(\displaystyle \begin{array}{r}\,\,\,\,3x-7\ge 41\,\,\,\\\text{Is}\,\,\,\,\,3(20)-7\ge 41?\\60-7\ge 41\,\,\,\\53\ge 41\,\,\,\\\text{Yes!}\,\,\,\,\,\end{array}\)

When solving multi-step equations, pay attention to situations in which you multiply or divide by a negative number. In these cases, you must reverse the inequality sign.

Solve for p . \(−58>14−6p\)

[reveal-answer q=”424351″]Show Solution[/reveal-answer] [hidden-answer a=”424351″]

Note how the variable is on the right hand side of the inequality, the method for solving does not change in this case.

Begin to isolate the variable by subtracting 14 from both sides of the inequality.

\(\displaystyle \begin{array}{l}−58\,\,>14−6p\\\underline{\,\,\,\,\,\,\,\,\,\,\,\,-14\,\,\,\,\,\,\,-14}\\-72\,\,\,\,\,\,\,\,\,\,\,>-6p\end{array}\)

Divide both sides of the inequality by \(−6\) to express the variable with a coefficient of 1. Dividing by a negative number results in reversing the inequality sign.

\(\begin{array}{l}\underline{-72}>\underline{-6p}\\-6\,\,\,\,\,\,\,\,\,\,-6\\\,\,\,\,\,\,12\lt{p}\end{array}\)

We can also write this as \(p>12\). Notice how the inequality sign is still opening up toward the variable p.

Inequality: \(\left(12,\infty\right)\) Graph: The graph of the inequality p > 12 has an open circle at 12 with an arrow stretching to the right.

Check the solution. [reveal-answer q=”500309″]Show Solution[/reveal-answer] [hidden-answer a=”500309″]

First, check the end point 12 in the related equation.

\(\begin{array}{r}-58=14-6p\\-58=14-6\left(12\right)\\-58=14-72\\-58=-58\end{array}\)

Then, try another value to check the inequality. Try 100.

\(\begin{array}{r}-58>14-6p\\-58>14-6\left(100\right)\\-58>14-600\\-58>-586\end{array}\)

In the following video, you will see an example of solving a linear inequality with the variable on the left side of the inequality, and an example of switching the direction of the inequality after dividing by a negative number.

In the following video, you will see an example of solving a linear inequality with the variable on the right side of the inequality, and an example of switching the direction of the inequality after dividing by a negative number.

Simplify and solve algebraic inequalities using the distributive property

As with equations, the distributive property can be applied to simplify expressions that are part of an inequality. Once the parentheses have been cleared, solving the inequality will be straightforward.

Solve for x . \(2\left(3x–5\right)\leq 4x+6\)

[reveal-answer q=”587737″]Show Solution[/reveal-answer] [hidden-answer a=”587737″]

Distribute to clear the parentheses.

\(\displaystyle \begin{array}{r}\,2(3x-5)\leq 4x+6\\\,\,\,\,6x-10\leq 4x+6\end{array}\)

Subtract 4 x from both sides to get the variable term on one side only.

\(\begin{array}{r}6x-10\le 4x+6\\\underline{-4x\,\,\,\,\,\,\,\,\,\,\,\,\,\,-4x}\,\,\,\,\,\,\,\,\,\\\,\,\,2x-10\,\,\leq \,\,\,\,\,\,\,\,\,\,\,\,6\end{array}\)

Add 10 to both sides to isolate the variable.

\(\begin{array}{r}\\\,\,\,2x-10\,\,\le \,\,\,\,\,\,\,\,6\,\,\,\\\underline{\,\,\,\,\,\,+10\,\,\,\,\,\,\,\,\,+10}\\\,\,\,2x\,\,\,\,\,\,\,\,\,\,\,\le \,\,\,\,\,16\,\,\,\end{array}\)

Divide both sides by 2 to express the variable with a coefficient of 1.

\(\begin{array}{r}\underline{2x}\le \,\,\,\underline{16}\\\,\,\,2\,\,\,\,\,\,\,\,\,\,\,\,\,2\,\,\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x\,\,\,\le \,\,\,\,\,8\end{array}\)

Inequality: \(\left(-\infty,8\right]\) Graph: The graph of this solution set includes 8 and everything left of 8 on the number line.

Check the solution. [reveal-answer q=”808701″]Show Solution[/reveal-answer] [hidden-answer a=”808701″]

First, check the end point 8 in the related equation.

\(\displaystyle \begin{array}{r}2(3x-5)=4x+6\,\,\,\,\,\,\\2(3\,\cdot \,8-5)=4\,\cdot \,8+6\\\,\,\,\,\,\,\,\,\,\,\,2(24-5)=32+6\,\,\,\,\,\,\\2(19)=38\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\\38=38\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\end{array}\)

Then, choose another solution and evaluate the inequality for that value to make sure it is a true statement. Try 0.

\(\displaystyle \begin{array}{l}2(3\,\cdot \,0-5)\le 4\,\cdot \,0+6?\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2(-5)\le 6\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-10\le 6\,\,\end{array}\)

\(\left(-\infty,8\right]\)

In the following video, you are given an example of how to solve a multi-step inequality that requires using the distributive property.

In the next example, you are given an inequality with a term that looks complicated. If you pause and think about how to use the order of operations to solve the inequality, it will hopefully seem like a straightforward problem. Use the textbox to write down what you think is the best first step to take.

Solve for a. \(\displaystyle\frac ParseError: EOF expected (click for details) Callstack: at (Courses/Lumen_Learning/Beginning_Algebra_(Lumen)/01:_Solving_Equations_and_Inequalities/1.05:_Solve_Inequalities), /content/body/div[7]/div/div/div[3]/div/p[2]/span[1], line 1, column 2 6{<2}\)

[practice-area rows=”1″][/practice-area]

[reveal-answer q=”701072″]Show Solution[/reveal-answer]

[hidden-answer a=”701072″]

Clear the fraction by multiplying both sides of the equation by 6.

\(\displaystyle \begin{array}{r}\frac ParseError: EOF expected (click for details) Callstack: at (Courses/Lumen_Learning/Beginning_Algebra_(Lumen)/01:_Solving_Equations_and_Inequalities/1.05:_Solve_Inequalities), /content/body/div[7]/div/div/div[3]/div/p[7]/span[1], line 1, column 2 6{<2}\,\,\,\,\,\,\,\,\\\\6\,\cdot \,\frac{2a-4}{6}<2\,\cdot \,6\\\\{2a-4}<12\,\,\,\,\,\,\end{array}\)

Add 4 to both sides to isolate the variable.

\(\displaystyle \begin{array}{r}2a-4<12\\\underline{\,\,\,+4\,\,\,\,+4}\\2a<16\end{array}\)

\(\displaystyle \begin{array}{c}\frac{2a}{2}<\,\frac{16}{2}\\\\a<8\end{array}\)

Inequality: \(a<8\)

Interval: \(\left(-\infty,8\right)\)

Check the solution. [reveal-answer q=”905072″]Show Solution[/reveal-answer]

[hidden-answer a=”905072″] First, check the end point 8 in the related equation.

\(\displaystyle \begin{array}{r}\frac{2a-4}{6}=2\,\,\,\,\\\\\text{Does}\,\,\,\frac{2(8)-4}{6}=2?\\\\\frac{16-4}{6}=2\,\,\,\,\\\\\frac{12}{6}=2\,\,\,\,\\\\2=2\,\,\,\,\\\\\text{Yes!}\,\,\,\,\,\end{array}\)

Then choose another solution and evaluate the inequality for that value to make sure it is a true statement. Try 5.

\(\displaystyle \begin{array}{r}\text{Is}\,\,\,\frac{2(5)-4}{6}<2?\\\\\frac{10-4}{6}<2\,\,\,\\\\\,\,\,\,\frac{6}{6}<2\,\,\,\\\\1<2\,\,\,\\\\\text{Yes!}\,\,\,\,\,\end{array}\)

Solving inequalities is very similar to solving equations, except you have to reverse the inequality symbols when you multiply or divide both sides of an inequality by a negative number. There are three ways to represent solutions to inequalities: an interval, a graph, and an inequality. Because there is usually more than one solution to an inequality, when you check your answer you should check the end point and one other value to check the direction of the inequality.

Inequalities can have a range of answers. The solutions are often graphed on a number line in order to visualize all of the solutions. Multi-step inequalities are solved using the same processes that work for solving equations with one exception. When you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality symbol. The inequality symbols stay the same whenever you add or subtract either positive or negative numbers to both sides of the inequality.

• Revision and Adaptation. Provided by : Lumen Learning. License : CC BY: Attribution
• Graph Linear Inequalities in One Variable (Basic). Authored by : James Sousa (Mathispower4u.com) for Lumen Learning. Located at : https://youtu.be/-kiAeGbSe5c . License : CC BY: Attribution
• Given Interval in Words, Graph and Give Inequality. Authored by : James Sousa (Mathispower4u.com) for Lumen Learning. Located at : https://youtu.be/E_ZWNVNEvOg . License : CC BY: Attribution
• Given an Inequality, Graph and Give Interval Notation. Authored by : James Sousa (Mathispower4u.com) for Lumen Learning. Located at : https://youtu.be/BKhDzNKjVBc . License : CC BY: Attribution
• Given Interval in Words, Graph and Give Interval Notation. Authored by : James Sousa (Mathispower4u.com) for Lumen Learning. Located at : https://youtu.be/OYkQ-McI2qg . License : CC BY: Attribution
• Given Interval Notation, Graph and Give Inequality. Authored by : James Sousa (Mathispower4u.com) for Lumen Learning. Located at : https://youtu.be/lkhILNEPbfk . License : CC BY: Attribution
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• Solutions to Basic OR Compound Inequalities. Authored by : James Sousa (Mathispower4u.com) for Lumen Learning. Located at : https://youtu.be/nKarzhZOFIk . License : CC BY: Attribution
• Solutions to Basic AND Compound Inequalities. Authored by : James Sousa (Mathispower4u.com) for Lumen Learning. Located at : https://youtu.be/LP3fsZNjJkc . License : CC BY: Attribution
• Unit 10: Solving Equations and Inequalities, from Developmental Math: An Open Program. Provided by : Monterey Institute of Technology and Education. Located at : nrocnetwork.org/resources/downloads/nroc-math-open-textbook-units-1-12-pdf-and-word-formats/. License : CC BY: Attribution
• Ex: Solving One Step Inequalities by Adding and Subtracting (Variable Left Side). Authored by : James Sousa (Mathispower4u.com) for Lumen Learning. Located at : https://youtu.be/1Z22Xh66VFM . License : CC BY: Attribution
• Ex: Solving One Step Inequalities by Adding and Subtracting (Variable Right Side). Authored by : James Sousa (Mathispower4u.com) for Lumen Learning. Located at : https://youtu.be/RBonYKvTCLU . License : CC BY: Attribution
• Ex: Solve One Step Linear Inequality by Dividing (Variable Left). Authored by : James Sousa (Mathispower4u.com) for Lumen Learning. Located at : https://youtu.be/IajiD3R7U-0 . License : CC BY: Attribution
• Ex: Solve One Step Linear Inequality by Dividing (Variable Right). Authored by : James Sousa (Mathispower4u.com) for Lumen Learning. Located at : https://youtu.be/s9fJOnVTHhs . License : CC BY: Attribution
• Ex: Graph Basic Inequalities and Express Using Interval Notation. Authored by : James Sousa (Mathispower4u.com) . Located at : https://youtu.be/X0xrHKgbDT0 . License : CC BY: Attribution
• College Algebra. Authored by : Jay Abramson, et al.. Located at : courses.candelalearning.com/collegealgebra1xmaster/. License : CC BY: Attribution
• Ex: Graph Basic Inequalities and Express Using Interval Notation. Authored by : James Sousa (Mathispower4u.com) for Lumen Learning. Located at : https://youtu.be/X0xrHKgbDT0 . License : CC BY: Attribution
• Ex: Solve a Two Step Linear Inequality (Variable Left). Authored by : James Sousa (Mathispower4u.com) for Lumen Learning. Located at : https://youtu.be/RB9wvIogoEM . License : CC BY: Attribution
• Ex: Solve a Two Step Linear Inequality (Variable Right). Authored by : James Sousa (Mathispower4u.com) . Located at : https://youtu.be/9D2g_FaNBkY . License : CC BY: Attribution
• Ex: Solve a Linear Inequality Requiring Multiple Steps (One Var). Authored by : James Sousa (Mathispower4u.com) for Lumen Learning. Located at : https://youtu.be/vjZ3rQFVkh8 . License : Public Domain: No Known Copyright
• Ex: Solve a Compound Inequality Involving OR (Union). Authored by : James Sousa (Mathispower4u.com) for Lumen Learning. Located at : https://youtu.be/oRlJ8G7trR8 . License : CC BY: Attribution
• Ex 1: Solve and Graph Basic Absolute Value inequalities. Authored by : James Sousa (Mathispower4u.com) for Lumen Learning. Located at : https://youtu.be/0cXxATY2S-k . License : CC BY: Attribution
• Ex 2: Solve and Graph Absolute Value inequalities Mathispower4u . Authored by : James Sousa (Mathispower4u.com) for Lumen Learning. Located at : https://youtu.be/d-hUviSkmqE . License : CC BY: Attribution
• Ex 3: Solve and Graph Absolute Value inequalitie. Authored by : James Sousa (Mathispower4u.com) . Located at : https://youtu.be/ttUaRf-GzpM . License : CC BY: Attribution
• Ex 4: Solve and Graph Absolute Value inequalities (Requires Isolating Abs. Value). Authored by : James Sousa (Mathispower4u.com) . Located at : https://youtu.be/5jRUuiMUxWQ . License : CC BY: Attribution
• College Algebra. Authored by : Jay Abramson, et. al. Located at : courses.candelalearning.com/collegealgebra1xmaster/. License : CC BY: Attribution

Inequalities

In Mathematics, equations are not always about being balanced on both sides with an 'equal to' symbol. Sometimes it can be about 'not an equal to' relationship like something is greater than the other or less than. In mathematics, inequality refers to a relationship that makes a non-equal comparison between two numbers or other mathematical expressions. These mathematical expressions come under algebra and are called inequalities.

Let us learn the rules of inequalities, and how to solve and graph them.

What is an Inequality?

Inequalities  are the mathematical expressions in which both sides are not equal. In inequality, unlike in equations, we compare two values. The equal sign in between is replaced by less than (or less than or equal to), greater than (or greater than or equal to), or not equal to sign.

Olivia is selected in the 12U Softball. How old is Olivia? You don't know the age of Olivia, because it doesn't say "equals". But you do know her age should be less than or equal to 12, so it can be written as Olivia's Age ≤ 12. This is a practical scenario related to inequalities.

Inequality Meaning

The meaning of inequality is to say that two things are NOT equal. One of the things may be less than, greater than, less than or equal to, or greater than or equal to the other things.

• p ≠ q means that p is not equal to q
• p < q means that p is less than q
• p > q means that p is greater than q
• p ≤ q means that p is less than or equal to q
• p ≥ q means that p is greater than or equal to q

There are different types of inequalities. Some of the important inequalities are:

• Polynomial inequalities
• Absolute value inequalities
• Rational inequalities

Rules of Inequalities

The rules of inequalities are special. Here are some listed with inequalities examples.

Inequalities Rule 1

When inequalities are linked up you can jump over the middle inequality.

• If, p < q and q < d, then p < d
• If, p > q and q > d, then p > d

Example: If Oggy is older than Mia and Mia is older than Cherry, then Oggy must be older than Cherry.

Inequalities Rule 2

Swapping of numbers p and q results in:

• If, p > q, then q < p
• If, p < q, then q > p

Example: Oggy is older than Mia, so Mia is younger than Oggy.

Inequalities Rule 3

Adding the number d to both sides of inequality: If p < q, then p + d < q + d

Example: Oggy has less money than Mia. If both Oggy and Mia get $5 more, then Oggy will still have less money than Mia.

• If p < q, then p − d < q − d
• If p > q, then p + d > q + d, and
• If p > q, then p − d > q − d

So, the addition and subtraction of the same value to both p and q will not change the inequality.

Inequalities Rule 4

If you multiply numbers p and q by a positive number , there is no change in inequality. If you multiply both p and q by a negative number , the inequality swaps: p<q becomes q<p after multiplying by (-2)

Here are the rules:

• If p < q, and d is positive, then pd < qd
• If p < q, and d is negative, then pd > qd (inequality swaps)

Positive case example: Oggy's score of 5 is lower than Mia's score of 9 (p < q). If Oggy and Mia double their scores '×2', Oggy's score will still be lower than Mia's score, 2p < 2q. If the scores turn minuses, then scores will be −p > −q.

Inequalities Rule 5

Putting minuses in front of p and q changes the direction of the inequality.

• If p < q then −p > −q
• If p > q, then −p < −q
• It is the same as multiplying by (-1) and changes direction.

Inequalities Rule 6

Taking the reciprocal 1/value of both p and q changes the direction of the inequality. When p and q are both positive or both negative:

• If, p < q, then 1/p > 1/q
• If p > q, then 1/p < 1/q

Inequalities Rule 7

A square of a number is always greater than or equal to zero p 2  ≥ 0.

Example: (4) 2 = 16, (−4) 2 = 16, (0) 2 = 0

Inequalities Rule 8

Taking a square root will not change the inequality. If p ≤ q, then √p ≤ √q (for p, q ≥ 0).

Example: p=2, q=7 2 ≤ 7, then √2 ≤ √7

The rules of inequalities are summarized in the following table.

Solving Inequalities

Here are the steps for  solving inequalities :

• Step - 1: Write the inequality as an equation.
• Step - 2: Solve the equation for one or more values.
• Step - 3: Represent all the values on the number line.
• Step - 4: Also, represent all excluded values on the number line using open circles.
• Step - 5: Identify the intervals.
• Step - 6: Take a random number from each interval, substitute it in the inequality and check whether the inequality is satisfied.
• Step - 7: Intervals that are satisfied are the solutions.

But for solving simple inequalities (linear), we usually apply algebraic operations like addition , subtraction , multiplication , and division . Consider the following example:

2x + 3 > 3x + 4

Subtracting 3x and 3 from both sides,

2x - 3x > 4 - 3

Multiplying both sides by -1,

Notice that we have changed the ">" symbol into "<" symbol. Why? This is because we have multiplied both sides of the inequality by a negative number. The process of solving inequalities mentioned above works for a simple linear inequality. But to solve any other complex inequality, we have to use the following process.

Let us use this procedure to solve inequalities of different types.

Graphing Inequalities

While graphing inequalities , we have to keep the following things in mind.

• If the endpoint is included (i.e., in case of ≤ or ≥) use a closed circle.
• If the endpoint is NOT included (i.e., in case of < or >), use an open circle.
• Use open circle at either ∞ or -∞.
• Draw a line from the endpoint that extends to the right side if the variable is greater than the number.
• Draw a line from the endpoint that extends to the left side if the variable is lesser than the number.

Writing Inequalities in Interval Notation

While writing the solution of an inequality in the interval notation , we have to keep the following things in mind.

• If the endpoint is included (i.e., in case of ≤ or ≥) use the closed brackets '[' or ']'
• If the endpoint is not included (i.e., in case of < or >), use the open brackets '(' or ')'
• Use always open bracket at either ∞ or -∞.

Here are some examples to understand the same:

Graphing Inequalities with Two Variables

For graphing  inequalities with two variables , you will have to plot the "equals" line and then, shade the appropriate area. There are three steps:

• Write the equation such as "y" is on the left and everything else on the right.
• Plot the "y=" line (draw a solid line for y≤ or y≥, and a dashed line for y< or y>)
• Shade the region above the line for a "greater than" (y> or y≥) or below the line for a "less than" (y< or y≤).

Let us try some example: This is a graph of a linear inequality: y ≤ x + 4

You can see, y = x + 4 line and the shaded area (in yellow) is where y is less than or equal to x + 4. Let us now see how to solve different types of inequalities and how to graph the solution in each case.

Solving Polynomial Inequalities

The polynomial inequalities are inequalities that can be expressed as a polynomial on one side and 0 on the other side of the inequality. There are different types of polynomial inequalities but the important ones are:

• Linear Inequalities
• Quadratic Inequalities

Solving Linear Inequalities

A linear inequality is an inequality that can be expressed with a linear expression on one side and a 0 on the other side. Solving linear inequalities is as same as solving linear equations , but just the rules of solving inequalities (that was explained before) should be taken care of. Let us see some examples.

Solving One Step Inequalities

Consider an inequality 2x < 6 (which is a linear inequality with one variable ). To solve this, just one step is sufficient which is dividing both sides by 2. Then we get x < 3. Therefore, the solution of the inequality is x < 3 (or) (-∞, 3).

Solving Two Step Inequalities

Consider an inequality -2x + 3 > 6. To solve this, we need two steps . The first step is subtracting 3 from both sides, which gives -2x > 3. Then we need to divide both sides by -2 and it results in x < -3/2 (note that we have changed the sign of the inequality). So the solution of the inequality is x < -3/2 (or) (-∞, -3/2).

Solving Compound Inequalities

Compound inequalities refer to the set of inequalities with either "and" or "or" in between them. For solving inequalities, in this case, just solve each inequality independently and then find the final solution according to the following rules:

• The final solution is the intersection of the solutions of the independent inequalities if we have "and" between them.
• The final solution is the union of the solutions of the independent inequalities if we have "or" between them.

Example: Solve the compound inequality 2x + 3 < -5 and x + 6 < 3.

By first inequality: 2x + 3 < -5 2x < -8 x < -4

By second inequality, x + 6 < 3 x < -3

Since we have "and" between them, we have to find the intersection of the sets x < -4 and x < -3. A number line may be helpful in this case. Then the final solution is:

x < -3 (or) (-∞, -3).

Solving Quadratic Inequalities

A quadratic inequality involves a quadratic expression in it. Here is the process of solving quadratic inequalities . The process is explained with an example where we are going to solve the inequality x 2 - 4x - 5 ≥ 0.

• Step 1: Write the inequality as equation. x 2 - 4x - 5 = 0
• Step 2: Solve the equation. Here we can use any process of solving quadratic equations . Then (x - 5) (x + 1) = 0 x = 5; x = -1.

• Step 5: The inequalities with "true" from the above table are solutions. Therefore, the solutions of the quadratic inequality x 2 - 4x - 5 ≥ 0 is (-∞, -1] U [5, ∞).

We can use the same process for solving cubic inequalities, biquadratic inequalities, etc.

Solving Absolute Value Inequalities

An absolute value inequality includes an algebraic expression inside the absolute value sign. Here is the process of solving absolute value inequalities where the process is explained with an example of solving an absolute value inequality |x + 3| ≤ 2. If you want to learn different methods of solving absolute value inequalities, click here .

• Step 1: Consider the absolute value inequality as equation. |x + 3| = 2
• Step 2: Solve the equation. x + 3 = ±2 x + 3 = 2; x + 3 = -2 x = -1; x = -5

• Step 5: The intervals that satisfied the inequality are the solution intervals. Therefore, the solution of the absolute value inequality |x + 3| ≤ 2 is [-5, -1].

Solving Rational Inequalities

Rational inequalities are inequalities that involve rational expressions (fractions with variables). To solve the rational inequalities (inequalities with fractions), we just use the same procedure as other inequalities but we have to take care of the excluded points . For example, while solving the rational inequality (x + 2) / (x - 2) < 3, we should note that the rational expression (x + 2) / (x - 2) is NOT defined at x = 2 (set the denominator x - 2 = 0 ⇒x = 2). Let us solve this inequality step by step.

• Step 1: Consider the inequality as the equation. (x + 2) / (x - 2) = 3
• Step 2: Solve the equation. x + 2 = 3(x - 2) x + 2 = 3x - 6 2x = 8 x = 4

• Step 5: The intervals that have come up with "true" in Step 4 are the solutions. Therefore, the solution of the rational inequality (x + 2) / (x - 2) < 3 is (-∞, 2) U (4, ∞).

Important Notes on Inequalities:

Here are the notes about inequalities:

• If we have strictly less than or strictly greater than symbol, then we never get any closed interval in the solution.
• We always get open intervals at ∞ or -∞ symbols because they are NOT numbers to include.
• Write open intervals always at excluded values when solving rational inequalities.
• Excluded values should be taken care of only in case of rational inequalities.

☛  Related Topics:

• Linear Inequalities Calculator
• Triangle Inequality Theorem Calculator
• Rational Inequalities Calculator

Inequalities Examples

Example 1: Using the techniques of solving inequalities, solve: -19 < 3x + 2 ≤ 17 and write the answer in the interval notation.

Given that -19 < 3x + 2 ≤ 17.

This is a compound inequality.

Subtracting 2 from all sides,

-21 < 3x ≤ 15

Dividing all sides by 3,

-7 < x ≤ 5

Answer: The solution is (-7, 5].

Example 2: While solving inequalities, explain why each of the following statements is incorrect. Also, correct them. a) 2x < 5 ⇒ x > 5/2 b) x > 3 ⇒ x ∈ [3, ∞) c) -x > -7 ⇒ x > 7.

a) 2x < 5. Here, when we divide both sides by 2, which is a positive number, the sign does not change. So the correct inequality is x < 5/2.

b) x > 3. It does not include an equal to symbol. So 3 should NOT be included in the interval. So the correct interval is (3, ∞).

c) -x > -7. When we divide both sides by -1, a negative number, the sign should change. So the correct inequality is x < 7.

Answer: The corrected ones are a) x < 5/2; b) x ∈ (3, ∞); c) x < 7.

Example 3: Solve the inequality x 2 - 7x + 10 < 0.

First, solve the equation x 2 - 7x + 10 = 0.

(x - 2) (x - 5) = 0.

x = 2, x = 5.

If we represent these numbers on the number line, we get the following intervals: (-∞, 2), (2, 5), and (5, ∞).

Let us take some random numbers from each interval to test the given quadratic inequality.

Therefore, the only interval that satisfies the inequality is (2, 5).

Answer: The solution is (2, 5).

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Practice Questions on Inequalities

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FAQs on Inequalities

What are inequalities in math.

When two or more algebraic expressions are compared using the symbols <, > ≤, or ≥, then they form an inequality. They   are the mathematical expressions in which both sides are not equal.

How Do you Solve Inequalities On A Number Line?

To plot an inequality in math, such as x>3, on a number line,

• Step 1: Draw a circle over the number (e.g., 3).
• Step 2: Check if the sign includes equal to (≥ or ≤) or not. If equal to sign is there along with > or <, then fill in the circle otherwise leave the circle unfilled.
• Step 3: On the number line, extend the line from 3(after encircling it) to show it is greater than or equal to 3.

How to Calculate Inequalities in Math?

To  calculate inequalities :

• just make it an equation
• mark the zeros  on the number line to get intervals
• test the intervals by taking any one number from it against the inequality.

Explain the Process of Solving Inequalities Graphically.

Solving inequalities graphically is possible when we have a system of two inequalities in two variables. In this case, we consider both inequalities as two linear equations and graph them. Then we get two lines. Shade the upper/lower portion of each of the lines that satisfies the inequality. The common portion of both shaded regions is the solution region.

What is the Difference Between Equations and Inequalities?

Here are the differences between equations and inequalities.

What Happens When you Square An Inequality?

A square of a number is always greater than or equal to zero p 2  ≥ 0. Example: (4) 2 = 16, (−4) 2 = 16, (0) 2 = 0

What are the Steps to Calculate Inequalities with Fractions?

Calculating inequalities with fractions is just like solving any other inequality. One easy way of solving such inequalities is to multiply every term on both sides by the LCD of all denominators so that all fractions become integers . For example, to solve (1/2) x + 1 > (3/4) x + 2, multiply both sides by 4. Then we get 2x + 4 > 3x + 8 ⇒ -x > 4 ⇒ x < -4.

What are the Steps for Solving Inequalities with Variables on Both Sides?

When an inequality has a variable on both sides, we have to try to isolate the variable. But in this process, flip the inequality sign whenever we are dividing or multiplying both sides by a negative number. Here is an example. 3x - 7 < 5x - 11 ⇒ -2x < -4 ⇒ x > 2.

How Do you Find the Range of Inequality?

You can find the range of values of x, by solving the inequality by considering it as a normal linear equation.

What Are the 5 Inequality Symbols?

The 5 inequality symbols are less than (<), greater than (>), less than or equal (≤), greater than or equal (≥), and the not equal symbol (≠).

How Do you Tell If It's An Inequality?

Equations and inequalities are mathematical sentences formed by relating two expressions to each other. In an equation, the two expressions are supposed to be equal and shown by the symbol =. Whereas in inequality, the two expressions are not necessarily equal and are indicated by the symbols: >, <, ≤ or ≥.

How to Graph the Solution After Solving Inequalities?

After solving inequalities, we can graph the solution keeping the following things in mind.

• Use an open circle at the number if it is not included and use a closed circle if it is included.
• Draw a line to the right side of the number in case of '>' and to the left side of the number in case of '<'.

Number Line

• -x+3\gt 2x+1
• (x+5)(x-5)\gt 0
• 2x^2-x\gt 0
• (x+3)^2\le 10x+6
• \left|3+2x\right|\le 7
• \frac{\left|3x+2\right|}{\left|x-1\right|}>2
• What are the 4 inequalities?
• There are four types of inequalities: greater than, less than, greater than or equal to, and less than or equal to.
• What is a inequality in math?
• In math, inequality represents the relative size or order of two values.
• How do you solve inequalities?
• To solve inequalities, isolate the variable on one side of the inequality, If you multiply or divide both sides by a negative number, flip the direction of the inequality.
• What are the 2 rules of inequalities?
• The two rules of inequalities are: If the same quantity is added to or subtracted from both sides of an inequality, the inequality remains true. If both sides of an inequality are multiplied or divided by the same positive quantity, the inequality remains true. If we multiply or divide both sides of an inequality by the same negative number, we must flip the direction of the inequality to maintain its truth.

inequalities-calculator

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What is Inequality in Math?

Inequality symbols, operations on inequalities, solving linear inequalities with addition, solving linear inequalities with subtraction, solving linear inequalities with multiplication, solving linear inequalities with division, solving linear inequalities using the distributive property, practice questions, solving inequalities – explanation & examples.

The word inequality means a mathematical expression in which the sides are not equal to each other. Basically, an inequality compares any two values and shows that one value is less than, greater than, or equal to the value on the other side of the equation.

Basically, there are five inequality symbols used to represent equations of inequality.

These inequality symbols are: less than ( < ), greater than ( > ), less than or equal ( ≤ ), greater than or equal ( ≥ ) and the not equal symbol ( ≠ ).

Inequalities are used to compare numbers and determine the range or ranges of values that satisfy the conditions of a given variable.

Operations on linear inequalities involve addition, subtraction, multiplication, and division. The general rules for these operations are shown below.

Although we have used < symbol for illustration, you should note that the same rules apply to >, ≤, and ≥.

• The inequality symbol does not change when the same number is added on both sides of the inequality. For example, if a< b, then a + c < b +
• Subtracting both sides of the inequality by the same number does not change the inequality sign. For example, if a< b, then a – c < b – c.
• Multiplying both sides of an inequality by a positive number does not change the inequality sign. For example, if a< b and if c is a positive number, then a * c < b *
• Dividing both sides of an inequality by a positive number does not change the inequality sign. If a< b and if c is a positive number, then a/c < b/c
• Multiplying both sides of an inequality equation by a negative number changes the direction of the inequality symbol. For example, given that a < b and c is a negative number, then a * c > b *

How to Solve Inequalities?

Like linear equations, inequalities can be solved by applying similar rules and steps with a few exceptions. The only difference when solving linear equations is an operation that involves multiplication or division by a negative number. Multiplying or dividing an inequality by a negative number changes the inequality symbol.

Linear inequalities can be solved using the following operations:

• Subtraction
• Multiplication
• Distribution of property

Let’s see a few examples below to understand this concept.

Solve 3x − 5 ≤ 3 − x.

We start by adding both sides of the inequality by 5

3x – 5 + 5 ≤ 3 + 5 − x

Then add both sides by x.

3x + x ≤ 8 – x + x

Finally, divide both sides of the inequality by 4 to get;

Calculate the range of values of y, which satisfies the inequality: y − 4 < 2y + 5.

Add both sides of the inequality by 4.

y – 4 + 4 < 2y + 5 + 4

y < 2y + 9

Subtract both sides by 2y.

y – 2y < 2y – 2y + 9

Y < 9 Multiply both sides of the inequality by −1 and change the inequality symbol’s direction. y > − 9

Solve x + 8 > 5.

Isolate the variable x by subtracting 8 from both sides of the inequality.

x + 8 – 8 > 5 – 8 => x > −3

Therefore, x > −3.

Solve 5x + 10 > 3x + 24.

Subtract 10 from both sides of the inequality.

5x + 10 – 10 > 3x + 24 – 10

5x > 3x + 14.

Now we subtract both sides of the inequality by 3x.

5x – 3x > 3x – 3x + 14

Solve x/4 > 5

Multiply both sides of an inequality by the denominator of the fraction

4(x/4) > 5 x 4

Solve -x/4 ≥ 10

Multiply both sides of an inequality by 4.

4(-x/4) ≥ 10 x 4

Multiply both sides of the inequality by -1 and reverse the direction of the inequality symbol.

x ≤ – 40

Solve the inequality: 8x − 2 > 0.

First of all, add both sides of the inequality by 2

8x – 2 + 2 > 0 + 2

Now, solve by dividing both sides of the inequality by 8 to get;

Solve the following inequality:

−5x > 100

Divide both sides of the inequality by -5 and change the direction of the inequality symbol

= −5x/-5 < 100/-5

= x < − 20

Solve: 2 (x – 4) ≥ 3x – 5

2 (x – 4) ≥ 3x – 5

Apply the distributive property to remove the parentheses.

⟹ 2x – 8 ≥ 3x – 5

Add both sides by 8.

⟹ 2x – 8 + 8 ≥ 3x – 5 + 8

⟹ 2x ≥ 3x + 3

Subtract both sides by 3.

⟹ 2x – 3x ≥ 3x + 3 – 3x

⟹ x ≤ – 3

A student scored 60 marks in the first test and 45 marks in the second test of the terminal examination. How many minimum marks should the student score in the third test get a mean of least 62 marks?

Let the marks scored in the third test be x marks.

(60 + 45 + x)/3 ≥ 62 105 + x ≥ 196 x ≥ 93 Therefore, the student must score 93 marks to maintain a mean of at least 62 marks.

Justin requires at least $500 to hold his birthday party. If already he has saved $ 150 and 7 months are left to this date. What is the minimum amount he must save monthly?

Let the minimum amount saved monthly = x

150 + 7x ≥ 500

Solve for x

150 – 150 + 7x ≥ 500 – 150

Therefore, Justin should save $50 or more

Find two consecutive odd numbers which are greater than 10 and have the sum of less than 40.

Let the smaller odd number = x

Therefore, the next number will be x + 2

x > 10 ………. greater than 10

x + (x + 2) < 40 ……sum is less 40

Solve the equations.

2x + 2 < 40

x + 1< 20

Combine the two expressions.

10 < x < 19

Therefore, the consecutive odd numbers are 11 and 13, 13 and 15, 15 and 17, 17 and 19.

Inequalities and the Number Line

The best tool to represent and visualize numbers is the number line. A number line is defined as a straight horizontal line with numbers placed along at equal segments or intervals. A number line has a neutral point at the middle, known as the origin. On the right side of the origin on the number line are positive numbers, while the left side of the origin is negative numbers.

Linear equations can also be solved by a graphical method using a number line. For example, to plot x > 1, on a number line, you circle the number 1 on the number line and draw a line going from the circle in the direction of the numbers that satisfies the inequality statement.

If the inequality symbol is greater than or equal to or less than or equal to sign (≥ or ≤), draw the circle over the numerical number and fill or shade the circle. Finally, draw a line going from the shaded circle in the numbers’ direction that satisfies the inequality equation.

The same procedure is used to solve equations involving intervals.

 Example 15

–2 <  x  < 2

–1 ≤  x  ≤ 2

–1 <  x  ≤ 2

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May 31, 2024

10 min read

Math Can Help Solve Social Justice Problems

Mathematicians are working on ways to use their field to tackle major social issues, such as social inequality and the need for gender equity

By Rachel Crowell & Nature magazine

MicroStockHub/Getty Images

When Carrie Diaz Eaton trained as a mathematician, they didn’t expect their career to involve social-justice research. Growing up in Providence, Rhode Island, Diaz Eaton first saw social justice in action when their father, who’s from Peru, helped other Spanish-speaking immigrants to settle in the United States.

But it would be decades before Diaz Eaton would forge a professional path to use their mathematical expertise to study social-justice issues. Eventually, after years of moving around for education and training, that journey brought them back to Providence, where they collaborated with the Woonasquatucket River Watershed Council on projects focused on preserving the local environment of the river’s drainage basin, and bolstering resources for the surrounding, often underserved communities.

By “thinking like a mathematician” and leaning on data analysis, data science and visualization skills, they found that their expertise was needed in surprising ways, says Diaz Eaton, who is now executive director of the Institute for a Racially Just, Inclusive, and Open STEM Education at Bates College in Lewiston, Maine.

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For example, the council identified a need to help local people to better connect with community resources. “Even though health care and education don’t seem to fall under the purview of a watershed council, these are all interrelated issues,” Diaz Eaton says. Air pollution can contribute to asthma attacks, for example. In one project, Diaz Eaton and their collaborators built a quiz to help community members to choose the right health-care option, depending on the nature of their illness or injury, immigration status and health-insurance coverage.

“One of the things that makes us mathematicians, is our skills in logic and the questioning of assumptions”, and creating that quiz “was an example of logic at play”, requiring a logic map of cases and all of the possible branches of decision-making to make an effective quiz, they say.

Maths might seem an unlikely bedfellow for social-justice research. But applying the rigour of the field is turning out to be a promising approach for identifying, and sometimes even implementing, fruitful solutions for social problems.

Mathematicians can experience first-hand the messiness and complexity — and satisfaction — of applying maths to problems that affect people and their communities. Trying to work out how to help people access much-needed resources, reduce violence in communities or boost gender equity requires different technical skills, ways of thinking and professional collaborations compared with breaking new ground in pure maths. Even for an applied mathematician like Diaz Eaton, transitioning to working on social-justice applications brings fresh challenges.

Mathematicians say that social-justice research is difficult yet fulfilling — these projects are worth taking on because of their tremendous potential for creating real-world solutions for people and the planet.

Data-driven research

Mathematicians are digging into issues that range from social inequality and health-care access to racial profiling and predictive policing. However, the scope of their research is limited by their access to the data, says Omayra Ortega, an applied mathematician and mathematical epidemiologist at Sonoma State University in Rohnert Park, California. “There has to be that measured information,” Ortega says.

Fortunately, data for social issues abound. “Our society is collecting data at a ridiculous pace,” Ortega notes. Her mathematical epidemiology work has examined which factors affect vaccine uptake in different communities. Her work has found, for example, that, in five years, a national rotavirus-vaccine programme in Egypt would reduce disease burden enough that the cost saving would offset 76% of the costs of the vaccine. “Whenever we’re talking about the distribution of resources, there’s that question of social justice: who gets the resources?” she says.

Lily Khadjavi’s journey with social-justice research began with an intriguing data set.

About 15 years ago, Khadjavi, a mathematician at Loyola Marymount University in Los Angeles, California, was “on the hunt for real-world data” for an undergraduate statistics class she was teaching. She wanted data that the students could crunch to “look at new information and pose their own questions”. She realized that Los Angeles Police Department (LAPD) traffic-stop data fit that description.

At that time, every time that LAPD officers stopped pedestrians or pulled over drivers, they were required to report stop data. Those data included “the perceived race or ethnicity of the person they had stopped”, Khadjavi notes.

When the students analysed the data, the results were memorable. “That was the first time I heard students do a computation absolutely correctly and then audibly gasp at their results,” she says. The data showed that one in every 5 or 6 police stops of Black male drivers resulted in a vehicle search — a rate that was more than triple the national average, which was about one out of every 20 stops for drivers of any race or ethnicity, says Khadjavi.

Her decision to incorporate that policing data into her class was a pivotal moment in Khadjavi’s career — it led to a key publication and years of building expertise in using maths to study racial profiling and police practice. She sits on California’s Racial Identity and Profiling Advisory Board , which makes policy recommendations to state and local agencies on how to eliminate racial profiling in law enforcement.

In 2023, she was awarded the Association for Women in Mathematics’ inaugural Mary & Alfie Gray Award for Social Justice, named after a mathematician couple who championed human rights and equity in maths and government.

Sometimes, gaining access to data is a matter of networking. One of Khadjavi’s colleagues shared Khadjavi’s pivotal article with specialists at the American Civil Liberties Union. In turn, these specialists shared key data obtained through public-records requests with Khadjavi and her colleague. “Getting access to that data really changed what we could analyse,” Khadjavi says. “[It] allowed us to shine a light on the experiences of civilians and police in hundreds of thousands of stops made every year in Los Angeles.”

The data-intensive nature of this research can be an adjustment for some mathematicians, requiring them to develop new skills and approach problems differently. Such was the case for Tian An Wong, a mathematician at the University of Michigan-Dearborn who trained in number theory and representation theory.

In 2020, Wong wanted to know more about the controversial issue of mathematicians collaborating with the police, which involves, in many cases, using mathematical modelling and data analysis to support policing activities. Some mathematicians were protesting about the practice as part of a larger wave of protests around systemic racism , following the killing of George Floyd by police in Minneapolis, Minnesota. Wong’s research led them to a technique called predictive policing, which Wong describes as “the use of historical crime and other data to predict where future crime will occur, and [to] allocate policing resources based on those predictions”.

Wong wanted to know whether the tactics that mathematicians use to support police work could instead be used to critique it. But first, they needed to gain some additional statistics and data analysis skills. To do so, Wong took an online introductory statistics course, re-familiarized themself with the Python programming language, and connected with colleagues trained in statistical methods. They also got used to reading research papers across several disciplines.

Currently, Wong applies those skills to investigating the policing effectiveness of a technology that automatically locates gunshots by sound. That technology has been deployed in parts of Detroit, Michigan, where community members and organizations have raised concerns about its multimillion-dollar cost and about whether such police surveillance makes a difference to public safety.

Getting the lay of the land

For some mathematicians, social-justice work is a natural extension of their career trajectories. “My choice of mathematical epidemiology was also partially born out of out of my love for social justice,” Ortega says. Mathematical epidemiologists apply maths to study disease occurrence in specific populations and how to mitigate disease spread. When Ortega’s PhD adviser mentioned that she could study the uptake of a then-new rotovirus vaccine in the mid-2000s, she was hooked.

Mathematicians, who decide to jump into studying social-justice issues anew, must do their homework and dedicate time to consider how best to collaborate with colleagues of diverse backgrounds.

Jonathan Dawes, an applied mathematician at the University of Bath, UK, investigates links between the United Nations’ Sustainable Development Goals (SDGs) and their associated target actions. Adopted in 2015, the SDGs are “a universal call to action to end poverty, protect the planet, and ensure that by 2030 all people enjoy peace and prosperity,” according to the United Nations , and each one has a number of targets.

“As a global agenda, it’s an invitation to everybody to get involved,” says Dawes. From a mathematical perspective, analysing connections in the complex system of SDGs “is a nice level of problem,” Dawes says. “You’ve got 17 Sustainable Development Goals. Between them, they have 169 targets. [That’s] an amount of data that isn’t very large in big-data terms, but just big enough that it’s quite hard to hold all of it in your head.”

Dawes’ interest in the SDGs was piqued when he read a 2015 review that focused on how making progress on individual goals could affect progress on the entire set. For instance, if progress is made on the goal to end poverty how does that affect progress on the goal to achieve quality education for all, as well as the other 15 SDGs?

“If there’s a network and you can put some numbers on the strengths and signs of the edges, then you’ve got a mathematized version of the problem,” Dawes says. Some of his results describe how the properties of the network change if one or more of the links is perturbed, much like an ecological food web. His work aims to identify hierarchies in the SDG networks, pinpointing which SDGs should be prioritized for the health of the entire system.

As Dawes dug into the SDGs, he realized that he needed to expand what he was reading to include different journals, including publications that were “written in very different ways”. That involved “trying to learn a new language”, he explains. He also kept up to date with the output of researchers and organizations doing important SDG-related work, such as the International Institute for Applied Systems Analysis in Laxenburg, Austria, and the Stockholm Environment Institute.

Dawes’ research showed that interactions between the SDGs mean that “there are lots of positive reinforcing effects between poverty, hunger, health care, education, gender equity and so on.” So, “it’s possible to lift all of those up” when progress is made on even one of the goals. With one exception: managing and protecting the oceans. Making progress on some of the other SDGs could, in some cases, stall progress for, or even harm, life below water.

Collaboration care

Because social-justice projects are often inherently cross-disciplinary, mathematicians studying social justice say it’s key in those cases to work with community leaders, activists or community members affected by the issues.

Getting acquainted with these stakeholders might not always feel comfortable or natural. For instance, when Dawes started his SDG research, he realized that he was entering a field in which researchers already knew each other, followed each other’s work and had decades of experience. “There’s a sense of being like an uninvited guest at a party,” Dawes says. He became more comfortable after talking with other researchers, who showed a genuine interest in what he brought to the discussion, and when his work was accepted by the field’s journals. Over time, he realized “the interdisciplinary space was big enough for all of us to contribute to”.

Even when mathematicians have been invited to join a team of social-justice researchers, they still must take care, because first impressions can set the tone.

Michael Small is an applied mathematician and director of the Data Institute at the University of Western Australia in Perth. For much of his career, Small focused on the behaviour of complex systems, or those with many simple interacting parts, and dynamical systems theory, which addresses physical and mechanical problems.

But when a former vice-chancellor at the university asked him whether he would meet with a group of psychiatrists and psychologists to discuss their research on mental health and suicide in young people, it transformed his research. After considering the potential social impact of better understanding the causes and risks of suicide in teenagers and younger children, and thinking about how the problem meshed well with his research in complex systems and ‘non-linear dynamics’, Small agreed to collaborate with the group.

The project has required Small to see beyond the numbers. For the children’s families, the young people are much more than a single data point. “If I go into the room [of mental-health professionals] just talking about mathematics, mathematics, mathematics, and how this is good because we can prove this really cool theorem, then I’m sure I will get push back,” he says. Instead, he notes, it’s important to be open to insights and potential solutions from other fields. Listening before talking can go a long way.

Small’s collaborative mindset has led him to other mental-health projects, such as the Transforming Indigenous Mental Health and Wellbeing project to establish culturally sensitive mental-health support for Indigenous Australians.

Career considerations

Mathematicians who engage in social-justice projects say that helping to create real-world change can be tremendously gratifying. Small wants “to work on problems that I think can do good” in the world. Spending time pursuing them “makes sense both as a technical challenge [and] as a social choice”, he says.

However, pursuing this line of maths research is not without career hurdles. “It can be very difficult to get [these kinds of] results published,” Small says. Although his university supports, and encourages, his mental-health research, most of his publications are related to his standard mathematics research. As such, he sees “a need for balance” between the two lines of research, because a paucity of publications can be a career deal breaker.

Diaz Eaton says that mathematicians pursuing social-justice research could experience varying degrees of support from their universities. “I’ve seen places where the work is supported, but it doesn’t count for tenure [or] it won’t help you on the job market,” they say.

Finding out whether social-justice research will be supported “is about having some really open and transparent conversations. Are the people who are going to write your recommendation letters going to see that work as scholarship?” Diaz Eaton notes.

All things considered, mathematicians should not feel daunted by wading into solving the world’s messy problems, Khadjavi says: “I would like people to follow their passions. It’s okay to start small.”

This article is reproduced with permission and was first published on May 22, 2024 .

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I love this app so much usually you have to pay a monthly fee but with this you don’t, I didn’t buy it at first. I thought that we would have to pay monthly and I didn’t feel like getting the app but I gave it out a try. After I gave it a few tries and math questions I realize it really does help and that I love it so much,being in middle school and having tough math helps me so much with this even though I don’t really read the videos on how to do it sure helps with the answers and that is what it’s great about this app because you can help me with with my homework and if I need help it’ll give me. I love this app so much definitely recommended you don’t have to pay any monthly fee and this is the first app. I’ve actually waited five stars and left a comment because usually these apps aren’t good, but this one is amazing. Love it so much. Thanks.

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I’ve never been so greatful towards an app before, usually there was a deal with other apps that you had only a few tries before you had to pay for the full thing or wait your time…which 90% of the time didn’t even give you an answer or help… I’ve never had something this smooth be so dang helpful 😭❤️ Not only did it help me pass my test (88% omg 😍) but it also gave helpful advice and dumbed it down for me so i didn’t have to read through everything. Quite literally gave me the information i wanted and also the answer which is AMAZING 😨❤️Helped me better understand the questions SO THANK YOU 🙏 will definitely be using this on math 🥶 hoping for the same smoothness on math since it’s what i struggle with but overall just from a first time use i’m deeply thankful for this app!! 😭🙏❤️

Thank you for your positive feedback! I'm delighted to hear that you're enjoying the application. Please feel free to share any suggestions you have directly, and I'll do my best to assist or pass them along to the appropriate team for consideration.

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iOS 18’s Big AI Update Could Automatically Solve Math Problems for You

Like predictive text, but for math equations.

WWDC 2024 hype season is upon us! We’re a little over a week away from Apple’s annual developer conference, where the company is expected to announce new versions of all of its platforms, including iOS 18 .

All rumors point to iOS 18 being the biggest update to the iPhone software in years, with generative AI as the headlining act. We’ve already heard of some obvious AI updates, like smart summaries, photo retouching, and automatic replies . However, a new report from AppleInsider suggests there’s even more AI in store. The Apple-focused publication says iOS 18 will get a “Catch Up” feature where Siri can provide an overview of recent notifications, cross-device media controls where you can activate Siri on one device to control another, and even the ability to create and edit images within iMessage using generative AI.

With all these leaked AI features, it might be easy to overlook perhaps the most underrated one: AI that can automatically solve math problems.

The Calculator functionality is going to be integrated into the Notes app.

No Need to Open Up the Calculator App

According to AppleInsider , a new feature called “Keyboard Math Predictions” will detect and automatically solve math equations that are entered as text. Think of this like your iPhone’s predictive text that finishes your sentences for you, but for math problems. On top of that, the Notes app will get a crash course in math, with the ability to recognize math equations and offer solutions with help from the Calculator app’s integration. Apple is also reportedly working on a way to generate graphs within the Notes app.

Apple’s AI isn’t the only one that can handle some math thrown its way. OpenAI previously announced GPT-4o, and the AI chatbot made easy work of an equation looking to solve for x. GPT-4o can even act as more of a personal tutor for more involved problems.

WWDC 2024 will finally reveal all the AI things that Apple has been working on.

Big AI Reveal on June 10

Fortunately, we only have to wait until Apple kicks off its WWDC 2024 on June 10 to see what’s in store with iOS 18. The software update won’t officially arrive until the fall, but Apple will share installable developer and public betas shortly after WWDC if you’re willing to deal with bugs and such throughout the summer.

With Apple hopping on the wave, generative AI chatbots are proving that they can handle math just as well as they can understand natural language. You could argue that AI-assisted math could make us all dumber, but that’s also what people said about calculators and computers. Sure, technology has made some people lazier, but it’s also helped save us time and solve even more complex problems. Adding AI that can solve math into the iPhone will only strengthen its purpose as do-it-all device.