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## Digital SAT Math

### Course: Digital SAT Math>Unit 2

Lesson 7: Linear inequality word problems: foundations

# Linear inequality word problems | Lesson

A guide to linear inequality word problems on the digital SAT

## What are linear inequality word problems?

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.
A system of linear inequalities is just like a system of linear equations, except it is composed of inequalities instead of equations.
Systems of linear inequalities are used to model scenarios with multiple constraints.
For example: You're buying snacks for a party; you want to buy enough so that you don't run out (snacks is greater than or equal to what people will eat), but you also don't want to overspend (money spent on snacks is less than or equal to budget for the party). If you manage to buy enough snacks without breaking your budget, you've solved a system of inequalities!
This lesson builds upon an understanding of the following skills:
• Solving linear equations and linear inequalities
• Understanding linear relationships
You can learn anything. Let's do this!

## How do I write linear inequalities based on word problems?

### Using inequalities to solve problems

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
minutes.

## How do I write systems of linear inequalities based on word problems?

### Translating systems of inequalities word problems

Writing systems of inequalities word problemSee video transcript

### Systems of linear inequalities word problems

On the SAT, systems of linear inequalities word problems are some of the longest questions you'll read. This can be intimidating, but don't worry—their bark is worse than their bite!
On the test, we may be asked to:
• Write our own system of linear inequalities based on the word problem
• Find a solution to the system we wrote

#### Let's look at some examples!

Diego works at a scooter dealership that sells two scooter models: a dollar sign, 5, comma, 000 standard model and a dollar sign, 7, comma, 000 racing model. Last month, his goal was to sell at least 36 scooters. If Diego met his goal and brought in over dollar sign, 250, comma, 000 in sales, which of the following systems of inequalities describes s, the possible number of standard model scooters, and r, the possible number of racing model scooters, that Diego sold last month?

Eugenia wants to buy at least 30 prizes for rewarding her students throughout the semester. The prize pool will be made of small and large prizes, which cost dollar sign, 2 and dollar sign, 5 each respectively. Her budget for the prizes can be no more than dollar sign, 100. She wants to buy at least 15 small prizes and at least 5 large prizes. Which of the following systems of inequalities represents the conditions described if x is the number of small prizes and y is the number of large prizes?
If Eugenia buys 10 large prizes, what is a possible number of small prizes she can buy to satisfy the conditions described?

### Try it!

Try: write a system of linear inequalities
A warehouse worker uses a forklift to move boxes that weigh either 45 pounds or 70 pounds each. Let x be the number of 45-pound boxes and y be the number of 70-pound boxes. The forklift can carry up to either 60 boxes or a weight of 3000 pounds.
The total number of boxes the forklift can carry is
60.
The total weight the forklift can carry is
3000 pounds.
Write an inequality that models the number of boxes the forklift can carry. The inequality should contain x, y, and an inequality sign.
Write an inequality that models the amount of weight the forklift can carry. The inequality should contain x, y, and an inequality sign.

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?

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?

Practice: write a system of linear inequalities
Diamond has two jobs. She works as a barista, which pays dollar sign, 13 per hour, and she works as an illustrator, which pays dollar sign, 30 per hour. She wants to work no more than 45 hours and to earn at least dollar sign, 625 per week. Her barista job schedules her for up to 30 hours per week. Which of the following systems of inequalities represents this situation in terms of x and y, where x is the number hours Diamond works as a barista and y is the number of hours she works as an illustrator?

Practice: solve a system of linear inequalities
A cargo van delivers only 50-pound and 80-pound packages. For each delivery trip, the van must carry at least 20 packages, and the total weight of the packages cannot exceed 1, comma, 500 pounds. What is the maximum number of 80-pound packages that the van can carry per trip?

## 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