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### 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\ge 30$$30$ minutes, $45$ minutes, etc.
"I wouldn't pay more than $\mathrm{}6$ for a sandwich!"$x\le 6$$\mathrm{}4$, $\mathrm{}5.50$, 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 $\ge$ what people will eat), but you also don't want to overspend (money spent on snacks $\le$ 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 $c$"$>c$
"Less than $c$" or "lower than $c$"$
"Greater than or equal to $c$" or "at least $c$"$\ge c$
"Less than or equal to $c$" or "at most $c$"$\le c$
"No less than $c$"$\ge c$
"No more than $c$"$\le c$
"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 $1.5$-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 $\mathrm{}0.40$ and a pickle costs $\mathrm{}2.30$, what is greatest number of peppermint stick pickles Bryan can make if he has $\mathrm{}20$ to buy the ingredients?

### Try it!

Try: solve a linear inequality word problem
Zoey has $\mathrm{}5$ and wants to rent a scooter. The scooter costs $\mathrm{}1$ to unlock and $\mathrm{}0.25$ 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.
$\text{total cost}=\phantom{\rule{0.167em}{0ex}}$
Since Zoey has $\mathrm{}5$, the cost of renting the scooter must be
$\mathrm{}5$.
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 $\mathrm{}5,000$ standard model and a $\mathrm{}7,000$ racing model. Last month, his goal was to sell at least $36$ scooters. If Diego met his goal and brought in over $\mathrm{}250,000$ 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 $\mathrm{}2$ and $\mathrm{}5$ each respectively. Her budget for the prizes can be no more than $\mathrm{}100$. 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 $\mathrm{}13$ per hour, and she works as an illustrator, which pays $\mathrm{}30$ per hour. She wants to work no more than $45$ hours and to earn at least $\mathrm{}625$ 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 $1,500$ 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 $c$"$>c$
"Less than $c$" or "lower than $c$"$
"Greater than or equal to $c$" or "at least $c$"$\ge c$
"Less than or equal to $c$" or "at most $c$"$\le c$
"No less than $c$"$\ge c$
"No more than $c$"$\le c$
"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

## Want to join the conversation?

• my brain isn't braining xD
• samee T__T
• I personally found this portion more challenging than any lesson. Any suggestion?
• exactly little hard
• peppermint stick pickles is crazy
• fr, I like pickles but this... this was deeply unsettling...
• I am struggling with these questions, any suggestions for how I can improve, I have my test on 3rd and these are just annoyingly confusing.
• just don't look at these charts. Imagine you are solving real problems. Then you will find it easy
• I used substitution method on the last question and got 16.67, and got it correct. So can i use substitution method even if they showed otherwise
• definitely, both methods work for any system of equations
• how did he take 10 and 30 in scooter question?
• many solutions to the question you can solve the question using the last lesson. The answer then would be 35 racing and 1 standard. 5,000(1)+7,000(35)=250,000
• The number of stops kayla can take in the first question should be 8 and not 7.6 as stated in the video , of this lesson, because when we approximate 7.6 to the nearest whole number, it becomes 8;that is, it is rounded up.
• True, but does she have that amount? Let's say you want to buy 5 apples that each costs a dollar. But you have \$4.80. Can you buy the 5th one? No, you'll just buy 4 apples.
• How do you solve this? am confused

5,000s + 7000r > 250,000
• Ok, so what are you trying to solve for? The variable s or r?
Let's just say you want to solve for s.
One thing to say before starting is that the steps for solving an inequality are the same as the steps for solving an equation except that if you multiply or divide by a negative number, you have to flip the inequality sign.
So, first to isolate s, I'm going to subtract 7000r from both sides. Now we have:
5000s > 250,000 - 7000r
Now to further isolate s, I'm going to divide both sides of the inequality by 5000. Since we're not dividing by a negative number, we don't have to flip the inequality sign. Here's what we got now:
s > 50 + 1.4r
We now know what s is! To find r, follow the same steps to isolate r on one side of the equation starting with the original equation.
Does that help at all?