equations and inequalities homework 8 word problems

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Writing Inequalities from Word Problems

Learn about writing inequalities from word problems with help from our practice examples. If you want to test yourself, or get some practice, then try one of our graded worksheets, or our online quiz.

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What is an Inequality?

The language of inequalities.

• How to Write an Inequality from a Word Problem
• Write an Inequality from a Word Problems Examples

Writing Inequalities from Word Problems Worksheets

• Writing Inequalities from Word Problems Online Quiz
• More related resources

An inequality is when you have a relationship between two values of expressions which are not equal to each other.

There are a few different options for different types of inequalities:

• Greater than (>) where one expression or value is greater than another, e.g. 7 > 5
• Less than (<) where one expression or value is less than another, e.g. 9 < 2 x 6
• Greater than or equal to (≥) where one expression or value is greater than or equal to another, e.g. 20 + 4 ≥ 17
• Less than or equal to (≤) where one expression or value is less than or equal to another, e.g. 18 ≤ 9 x 2
• Not equal to (≠) where one expression or value is not equal to another, e.g. 7 ≠ 4

When writing inequalities from word problems, we have to look carefully at and understand the language being used.

Different words and phrases have different meanings when deciding on which inequality to use.

The mathematical notation is really just a shorthand way of writing the words more efficiently and clearly.

Here is a quick table showing some of the written expressions often used and which inequality they are represented by.

Note: the word 'between' is mainly used to mean between inclusively (including end points).

However, sometimes 'between' is used to mean between exclusively (excluding end points).

To avoid ambiguity, it is good practice to include the word 'inclusive' or 'exclusive' to make it completely clear if the end points are included or not.

Some simple examples showing inequalities from phrases:

The variable names (letters) have been chosen at random - you can use any variable name to represent any value.

Note: you need to read the word problem carefully because sometimes the inequality does not match the language used, especially when the inequality involves finding out what is left over or what remains after an amount is taken away. See Examples 2) and 7) below.

How to Write Inequalities from Word Problems

When we are writing an inequality from a word problem, we are basically translating the word problem into mathematical language and symbols.

When writing an inequality from a word problem, there are two simple steps you need to follow...

Step 1) Read the word problem carefully and change the word problem into algebra.

• use the language of inequalities table to help you select the right inequality

Step 2) Use algebra to solve the word problem

Step 3) rewrite the inequality using algebra., write an inequality from a word problem examples.

The best way to learn how to write inequalities from word problems and see how they work is to look at some ready made examples.

Writing Inequalities from Word Problems - Basic Examples

Here are some examples of writing inequalities from word problems.

Example 1) Sally bakes some cookies and needs to put them in the over for at least 12 minutes. Write an inequality using the variable t to show how long the cookies need to be baked in the oven.

The vocabulary which tells us about the inequality are the words: at least .

This means we need to use the ≥ symbol.

So the inequality is t ≥ 12 minutes

Example 2) Newton has a 30 ounce bottle of water. He drinks over half of the bottle. Write an inequality using the variable c to show how many ounces are left in the bottle.

The vocabulary which tells us about the inequality are the words: over .

However, because he has drunk over half the bottle, it means that there is under half a bottle left.

So the symbol we need is < and the amount is ½ of 30 = 15.

So the inequality is b < 15 ounces

Example 3) Anna is more than three times as old as Bertie. If Bertie is 8 years old, write an inequality using the variable A to show how old Anna is.

The vocabulary which tells us about the inequality are the words: more than .

So the symbol we need is > and the amount is 3 x 8 = 24.

So the inequality is A > 24 years old

Example 4) A book has 14 chapters.The shortest chapter has 12 pages. Write an inequality using the variable p to show how many pages the book has.

The vocabulary which tells us about the inequality are the words: shortest .

If the shortest chapter has 11 pages, then there must be some chapters with more than 11 pages.

So the symbol we need is > and the amount is 14 x 12 = 168.

So the inequality is p > 168 pages.

Writing Inequalities from Word Problems -Intermediate Examples

These examples use two different variables and express one variable in terms of another.

Example 5) Captain and Frazer have some gold coins. Captain has at least three times as many coins as Frazer. Write an inequality for the number of coins Captain has (c) in terms of the number of coins Frazer has (f).

So the symbol we need is ≥

So the inequality is c ≥ 3f.

Example 6) In a hotel there are f flights of stairs. Each flight has a maximum of 12 steps. There are also 3 steps up to the main entrance. Write an expression for the total number of steps, s, in terms of f.

The vocabulary which tells us about the inequality are the words: a maximum of .

So the symbol we need is ≤

We know that there are f flights of steps and also 3 extra steps.

So the inequality is s ≤ 12f + 3.

Writing Inequalities from Word Problems - Harder Examples

These examples involve solving word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers.

There are also examples where the variable lies between two values.

Example 7) Captain has a one-liter bottle of water. He drinks more than one-quarter of the bottle but less than one-half of the bottle. Write an inequality using the variable b to show the amount of water than is left in the bottle.

The vocabulary which tells us about the inequality are the words: more than and less than .

However, because we are looking at what is left in the bottle, rather than what has been drunk, we need to think carefully about the inequalities!

He drinks more than one-quarter of the bottle, so there will be less than three-quarters of the bottle left, so we need the symbol <

He drinks less than one-half of the bottle, so there will be one-half or more of the bottle left, so we need the symbol ≥

Half of the bottle = ½ liters = 500 ml. 1000 - 500 = 500 ml

Quarter of the bottle = ¼ liters = 250 ml. 1000 - 250 = 750 ml

So the inequality is b ≥ 500 ml and b < 750 ml This can be simplified to: 500 ≤ b < 750 ml This means that he has at least 500 ml but less than 750 ml left.

Example 8) Captain has challenged himself to catch a minimum of 50 fish from a lake. He manages to catch 8 of them and put them in his bucket. If he catches 6 fish every hour, write an inequality to show the time (t) in hours it will take him to reach his target.

The vocabulary which tells us about the inequality are the words: a minimum of .

The inequality we get from this problem is 6t + 8 ≥ 50

We are not finished yet, because this needs to be simplified and written in terms of t.

6t + 8 ≥ 50 so 6t ≥ 42

If we divide both sides of this inequality by 6, we get:

So the inequality is t ≥ 7 hours He needs to fish for at least 7 hours to reach his target.

Example 9) It takes Newton between 23 and 28 seconds (inclusive) to swim a length of a swimming pool. Write an inequality using the variable t to show how long it will take him to swim 3 lengths.

The vocabulary which tells us about the inequality are the words: between (inclusive) .

So the symbol we need is ≤ and ≥

3 x 23 = 69 and 3 x 28 = 84

So the inequality is t ≥ 69 and t ≤ 84 This can be simplified to: 69 ≤ t ≤ 84 It will take him between 69 and 84 seconds (inclusive) to swim 3 lengths.

We have a range of different inequality worksheets which involve writing inequalities from a range of word problems..

We have split the sheets into 3 sections: A, B and C

• Section A involves basic level questions aimed at 6th grade
• Section B involves medium level questions aimed at 6th and 7th grade
• Section C involves more advanced questions aimed at 7th and 8th grade

Writing Inequalities from Word Problems - Section A Easier

Sheet 1 involves picking the vocabulary and relevant information from the problem and writing the inequality

Sheet 2 involves the same skills as Sheet 1, but also involves an arithmetic operation to get the inequality.

• Inequalities from Word Problems Sheet A1
• PDF version
• Inequalities from Word Problems Sheet A2

Writing Inequalities from Word Problems - Section B Medium

Sheet 1 involves using two variables and writing an inequality for one variable in terms of the other variable

Sheet 2 is similar to Sheet 1 but with slightly harder problems.

• Inequalities from Word Problems Sheet B1
• Inequalities from Word Problems Sheet B2

Writing Inequalities from Word Problems - Section C

Sheet 1 involves using one variables and using the information to solve the inequality, usually in the form px + q > r or px + q < r, where p, q, and r are specific rational numbers

Sheet 2 involves the same skills as Sheet 1 but has compound inequalities in each question

• Inequalities from Word Problems Sheet C1
• Inequalities from Word Problems Sheet C2

More Recommended Math Worksheets

Take a look at some more of our worksheets similar to these.

Inequalities on a Number Line Worksheets

• Inequalities on a Number Line

6th Grade Ratio and Unit Rate Worksheets

These 5th grade ratio worksheets are a great way to introduce this concept.

We have a range of part to part ratio worksheets and slightly harder problem solving worksheets.

• Ratio Part to Part Worksheets
• Ratio and Proportion Worksheets
• The Definition of Unit Rate
• Unit Rate Problems 6th Grade

6th Grade Algebra Worksheets

If you are looking for some 6th grade algebra worksheets to use with your child to help them understand simple equations then try our selection of basic algebra worksheets.

There are a range of 6th grade math worksheets covering the following concepts:

• Generate the algebra - and write your own algebraic expressions;
• Calculate the algebra - work out the value of different expressions;
• Solve the algebra - find the value of the term in the equation.
• Use the distributive property to factorize and expand different expressions
• 6th Grade Distributive Property Worksheets
• Expressions and Equations 6th Grade
• Basic Algebra Worksheets (6th & 7th Grade)

Writing Inequalities from Word Problems Quiz

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This quick quiz tests your skill at writing inequalities from a range of word problems.

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Equation Word Problems Worksheets

This compilation of a meticulously drafted equation word problems worksheets is designed to get students to write and solve a variety of one-step, two-step and multi-step equations that involve integers, fractions, and decimals. These worksheets are best suited for students in grade 6 through high school. Click on the 'Free' icons to sample our handouts.

One Step Equation Word Problem Worksheets

Read and solve this series of word problems that involve one-step equations. Apply basic operations to find the value of unknowns.

(15 Worksheets)

Two-Step Equation Word Problems: Integers

Interpret this set of word problems that require two-step operations to solve the equations. Each printable worksheet has five word problems ideal for 6th grade, 7th grade, and 8th grade students.

• Download the set

Multi-Step Equation Word Problems: Integers

Read each multi-step word problem in these high school pdf worksheets and set up the equation. Solve and find the value of the unknown. More than two steps are required to solve the problems.

Two-Step Equation Word Problems: Fractions and Decimals

Read each word problem and set up the two-step equation. Solve the equation and find the solution. This selection of worksheets includes both fractions and decimals.

Multi-Step Equation Word Problems: Fractions and Decimals

Write multi-step equations that involve both fractions and decimals based on the word problems provided here. Validate your responses with our answer keys.

MCQ - Equation Word Problems

Pick the correct two-step equation that best matches word problems presented here. Evaluate the ability of students to solve two-step equations with this array of MCQ worksheets.

Related Worksheets

» One-Step Equation

» Two-Step Equation

» Multi-Step Equation

» Algebraic Identities

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Equations and Inequalities Word Problems Worksheets

Inequalities are mathematically comparisons of values as where equations are all about displaying equality. Equations are math statements that shows two equal expressions. Inequalities state that one side of the math statement is larger or smaller than the other. These worksheets and lessons teach students how to write equations and inequalities that are based on word problems. This will empower your students to solve much more complex problem types that will require critical thinking skills that they should be ready for at the high school level. The goal is for them to be able to read these word problems and represent them with mathematical statements of their own.

Aligned Standard: HSA-CED.A.3

• Simple Equations Step-by-step Lesson - Who ever knew that picking a dance class could be such a mathematically dilemma?
• Guided Lesson - The problems are a bit lengthy to help you work with student and teach them to outline.
• Guided Lesson Explanation - Graphing them really helps bring the answers to you quickly.
• Practice Worksheet - I found these to be some of the most difficulty word problems to write in my life. To vary all the scenarios is quite difficult.
• Matching Worksheet - Match every heavy worded problem to it's output.
• Answer Keys - These are for all the unlocked materials above.

Homework Sheets

This section took me forever to write. I just kept running out of ideas.

• Homework 1 - Jian wants to take part in a Taekwondo class. He has two choices. Taekwondo class A costs $10 per month and $100 as registration fees.
• Homework 2 - Wilson is planning to open a Yoga studio. He has to pay $30 as rent for the studio and $5 per member for utilities. Each member will pay $15 for the yoga class.
• Homework 3 - George makes an average 15 baskets per game and made 40 baskets so far. Shane makes an average 10 baskets per game and made 55 baskets so far. If their averages remain the same, how many games do they have to play in order to score the same number of baskets?

Practice Worksheets

A big thanks to all my past colleagues that throw ideas at me to come up with new problems on this one.

• Practice 1 - Daniel has an office. He has to pay $20 a day in rent for the office and $2 per hour for parking. He pays $30 each day. How many hours does Daniel park his car?
• Practice 2 - Emma organized a party in a club. She has to pay $200 to an event organizer and her expenses for dinner are $10 per person. Each person pays $15 to Emma. How many people does Emma need to come to the party to reach a breakeven point?
• Practice 3 - Eva wants to print copies of her notes in hardcover book format. She has two choices. Printing A costs a setup fee of $40 and $3 for every book. Printing B costs a setup fee of $30 and $4 for every book. Find out when the two options cost the same amount. How many books is that?

Math Skill Quizzes

This can be a challenging task for many students.

• Quiz 1 - Two road crews are paving a straight section of road. The first crew has completed paving 12 miles and is completing 2 additional miles per day. The second crew has finished 8 miles and is completing 3 miles per day. How long will it take for both teams to cover the same number of miles?
• Quiz 2 - Vijay was renting a skateboard for his competition training. Skateboard A is $12 per month and requires a down payment of $100. Skateboard B is $15 per month rent and requires a down payment of $80. How many months would Vijay need to rent the skateboard in order for the total cost to be same?
• Quiz 3 - Marlin and Watson are playing video games. Marlin has 40 points and is earning 5 points every turn. Watson has 50 points and is earning 3 points every turn. In a certain number of turns, the score will be tied. How many points will they each have?

How to Write a System of Equations

When you are solving real-world algebraic equations, you can get easily confused. Considering that such problems are complex to solve and are known as the system of equations. Let us consider an example to learn its concept quickly. David spent $131 one pair of shoes, since one pair of sneakers had a price of $15, while the slippers had a cost of $28. In total, he bought 7 pairs of shoes. How many pairs of each shoe did he buy of each type?

In this section, we are only learning how to write the system of equations and not solve them. In this problem, we have to write two equations, given that there are two variables. The two variables are the two pairs of shoes, i.e., the sneakers and the slippers. Let us consider the two variables like x and y. Now, in total, David bought 7 pairs of shoes comprising of the two variables. So, the first equation becomes:

[1] => x + y = 7

The first equation depicts the number of shoes bought by David. The second equation will illustrate the total money he spent. He spent a total of $131 comprising of $15 worth of x’s and $28 worth of y’s. So, the equation becomes:

[2] => 15x + 28y = 7

Now, you have two equations that you need to simultaneously solve to find out how many pairs did David bought for each type of shoes.

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Inequalities Word Problems Worksheets

Inequalities word problems worksheets can help encourage students to read and think about the questions, rather than simply recognizing a pattern to the solutions.Inequalities word problems worksheet come with the answer key and detailed solutions which the students can refer to anytime.

Benefits of Inequalities Word Problems Worksheets

Inequalities word problems worksheets help kids to improve their speed, accuracy, logical and reasoning skills.

Inequalities word problems worksheets gives students the opportunity to solve a wide variety of problems helping them to build a robust mathematical foundation. Inequalities word problems worksheets helps kids to improve their speed, accuracy, logical and reasoning skills in performing simple calculations related to the topic of inequalities.

Inequalities word problems worksheets are also helpful for students to prepare for various competitive exams.

These worksheets come with visual simulation for students to see the problems in action, and provides a detailed step-by-step solution for students to understand the process better, and a worksheet properly explained about the Inequalities.

Download Inequalities Word Problems Worksheet PDFs

These math worksheets should be practiced regularly and are free to download in PDF formats.

☛ Check Grade wise Inequalities Word Problems Worksheets

• 7th Grade Inequalities Worksheets

Solving Inequality Word Questions

(You might like to read Introduction to Inequalities and Solving Inequalities first.)

In Algebra we have "inequality" questions like:

Sam and Alex play in the same soccer team. Last Saturday Alex scored 3 more goals than Sam, but together they scored less than 9 goals. What are the possible number of goals Alex scored?

How do we solve them?

The trick is to break the solution into two parts:

Turn the English into Algebra.

Then use Algebra to solve.

Turning English into Algebra

To turn the English into Algebra it helps to:

• Read the whole thing first
• Do a sketch if needed
• Assign letters for the values
• Find or work out formulas

We should also write down what is actually being asked for , so we know where we are going and when we have arrived!

The best way to learn this is by example, so let's try our first example:

Assign Letters:

• the number of goals Alex scored: A
• the number of goals Sam scored: S

We know that Alex scored 3 more goals than Sam did, so: A = S + 3

And we know that together they scored less than 9 goals: S + A < 9

We are being asked for how many goals Alex might have scored: A

Sam scored less than 3 goals, which means that Sam could have scored 0, 1 or 2 goals.

Alex scored 3 more goals than Sam did, so Alex could have scored 3, 4, or 5 goals .

• When S = 0, then A = 3 and S + A = 3, and 3 < 9 is correct
• When S = 1, then A = 4 and S + A = 5, and 5 < 9 is correct
• When S = 2, then A = 5 and S + A = 7, and 7 < 9 is correct
• (But when S = 3, then A = 6 and S + A = 9, and 9 < 9 is incorrect)

Lots More Examples!

Example: Of 8 pups, there are more girls than boys. How many girl pups could there be?

• the number of girls: g
• the number of boys: b

We know that there are 8 pups, so: g + b = 8, which can be rearranged to

We also know there are more girls than boys, so:

We are being asked for the number of girl pups: g

So there could be 5, 6, 7 or 8 girl pups.

Could there be 8 girl pups? Then there would be no boys at all, and the question isn't clear on that point (sometimes questions are like that).

• When g = 8, then b = 0 and g > b is correct (but is b = 0 allowed?)
• When g = 7, then b = 1 and g > b is correct
• When g = 6, then b = 2 and g > b is correct
• When g = 5, then b = 3 and g > b is correct
• (But if g = 4, then b = 4 and g > b is incorrect)

A speedy example:

Example: Joe enters a race where he has to cycle and run. He cycles a distance of 25 km, and then runs for 20 km. His average running speed is half of his average cycling speed. Joe completes the race in less than 2½ hours, what can we say about his average speeds?

• Average running speed: s
• So average cycling speed: 2s
• Speed = Distance Time
• Which can be rearranged to: Time = Distance Speed

We are being asked for his average speeds: s and 2s

The race is divided into two parts:

• Distance = 25 km
• Average speed = 2s km/h
• So Time = Distance Average Speed = 25 2s hours
• Distance = 20 km
• Average speed = s km/h
• So Time = Distance Average Speed = 20 s hours

Joe completes the race in less than 2½ hours

• The total time < 2½
• 25 2s + 20 s < 2½

So his average speed running is greater than 13 km/h and his average speed cycling is greater than 26 km/h

In this example we get to use two inequalities at once:

Example: The velocity v m/s of a ball thrown directly up in the air is given by v = 20 − 10t , where t is the time in seconds. At what times will the velocity be between 10 m/s and 15 m/s?

• velocity in m/s: v
• the time in seconds: t
• v = 20 − 10t

We are being asked for the time t when v is between 5 and 15 m/s:

So the velocity is between 10 m/s and 15 m/s between 0.5 and 1 second after.

And a reasonably hard example to finish with:

Example: A rectangular room fits at least 7 tables that each have 1 square meter of surface area. The perimeter of the room is 16 m. What could the width and length of the room be?

Make a sketch: we don't know the size of the tables, only their area, they may fit perfectly or not!

• the length of the room: L
• the width of the room: W

The formula for the perimeter is 2(W + L) , and we know it is 16 m

• 2(W + L) = 16
• L = 8 − W

We also know the area of a rectangle is the width times the length: Area = W × L

And the area must be greater than or equal to 7:

• W × L ≥ 7

We are being asked for the possible values of W and L

Let's solve:

So the width must be between 1 m and 7 m (inclusive) and the length is 8−width .

• Say W = 1, then L = 8−1 = 7, and A = 1 x 7 = 7 m 2 (fits exactly 7 tables)
• Say W = 0.9 (less than 1), then L = 7.1, and A = 0.9 x 7.1 = 6.39 m 2 (7 won't fit)
• Say W = 1.1 (just above 1), then L = 6.9, and A = 1.1 x 6.9 = 7.59 m 2 (7 fit easily)
• Likewise for W around 7 m

Inequalities Practice Questions

Click here for questions, click here for answers, gcse revision cards.

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Unit Two Equations And Inequalities Homework 8 Word Problems

Displaying top 8 worksheets found for - Unit Two Equations And Inequalities Homework 8 Word Problems .

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1. Unit 2 Reasoning With Equations And Inequalities Answers

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Unit 1: Algebra foundations

Unit 2: solving equations & inequalities, unit 3: working with units, unit 4: linear equations & graphs, unit 5: forms of linear equations, unit 6: systems of equations, unit 7: inequalities (systems & graphs), unit 8: functions, unit 9: sequences, unit 10: absolute value & piecewise functions, unit 11: exponents & radicals, unit 12: exponential growth & decay, unit 13: quadratics: multiplying & factoring, unit 14: quadratic functions & equations, unit 15: irrational numbers, unit 16: creativity in algebra.