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SAT

Course: SAT > Unit 6

Lesson 3: Heart of Algebra: lessons by skill
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Heart of Algebra: lessons by skill
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Linear inequality word problems | Lesson

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What are linear inequality word problems, and how frequently do they appear on the test?

Linear inequalities are very common in everyday life. While a linear equation gives us exactly one value when solved, a linear inequality gives us multiple values. The table below shows a couple of statements, their inequalities, and possible solutions.
StatementInequalityPossible solutions
"It'll take at least 30 minutes to get downtown."x, is greater than or equal to, 3030 minutes, 45 minutes, etc.
"I wouldn't pay more than dollar sign, 6 for a sandwich!"x, is less than or equal to, 6dollar sign, 4, dollar sign, 5, point, 50, etc.
On your official SAT, you'll likely see 1 to 2 questions that test your ability to write and solve linear inequalities.
This lesson builds upon an understanding of the following skills:
You can learn anything. Let's do this!

How do I write linear inequalities based on word problems?

Using inequalities to solve problems

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Using inequalities to solve problemsSee video transcript

Linear inequality word problems

It may not be hard to translate "it takes at least 30 minutes to get downtown" into a linear inequality, but some SAT word problems are several sentences long, and the information we need to build an inequality may be scattered around.

What are some key phrases to look out for?

The table below lists some common key phrases in inequality word problems and how to interpret them.
Note: c is a constant in the examples.
PhraseTranslates to...
"More than c", "greater than c", or "higher than c"is greater than, c
"Less than c" or "lower than c"is less than, c
"Greater than or equal to c" or "at least c"is greater than or equal to, c
"Less than or equal to c" or "at most c"is less than or equal to, c
"No less than c"is greater than or equal to, c
"No more than c"is less than or equal to, c
"Least", "lowest", or "minimum" valueThe smallest value that satisfies the inequality
"Greatest", "highest", or "maximum" valueThe largest value that satisfies the inequality
"A possible" valueAny value that satisfies the inequality

Let's look at some examples!

Ari can harvest at least 48 pounds of honey from her bee colony. If she wants to package the honey harvest in 1, point, 5-pound jars, what is the minimum number of jars she can fill?
Bryan wants to make
for his friends. The snack is made by inserting a peppermint stick into the middle of a pickle. If a peppermint stick costs dollar sign, 0, point, 40 and a pickle costs dollar sign, 2, point, 30, what is greatest number of peppermint stick pickles Bryan can make if he has dollar sign, 20 to buy the ingredients?

Try it!

Try: solve a linear inequality word problem
Zoey has dollar sign, 5 and wants to rent a scooter. The scooter costs dollar sign, 1 to unlock and dollar sign, 0, point, 25 for each minute of use.
Write an expression for the total cost in dollars of renting the scooter if Zoey uses it for x minutes.
start text, t, o, t, a, l, space, c, o, s, t, end text, equals
Since Zoey has dollar sign, 5, the cost of renting the scooter must be
dollar sign, 5.
If Zoey rents the scooter, she can use it for at most
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
minutes.

Your turn!

Practice: write a linear inequality
Cristian's goal is to walk an average of 50 kilometers a week for 4 weeks. He walked 56 kilometers the first week, 52 kilometers the second week, and 38 kilometers the third week. Which inequality can be used to represent the number of kilometers, x, Cristian could walk on the fourth week to meet his goal?
Choose 1 answer:

Practice: solve a linear inequality word problem
Dinah is driving on the highway. She must drive at a speed of at least 60 miles per hour and at most 70 miles per hour. Based on this information, what is a possible amount of time, in hours, that it could take Dinah to drive 420 miles?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

Things to remember

PhraseTranslates to...
"More than c", "greater than c", or "higher than c"is greater than, c
"Less than c" or "lower than c"is less than, c
"Greater than or equal to c" or "at least c"is greater than or equal to, c
"Less than or equal to c" or "at most c"is less than or equal to, c
"No less than c"is greater than or equal to, c
"No more than c"is less than or equal to, c
"Least", "lowest", or "minimum" valueThe smallest value that satisfies the inequality
"Greatest", "highest", or "maximum" valueThe largest value that satisfies the inequality
"A possible" valueAny value that satisfies the inequality

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