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

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

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

Guys please any help how to prepare? i have test on 9th of march

I've got a test on the 9th too. All I could advise is to finish the Khan Math Course, perfect your weaknesses, and take practice tests on Bluebook.

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?

The number of stops kayla can take in the first question should be 8 and not 7.6 as stated in the video ,3:18 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.

Main content

Course: Digital SAT Math > Unit 2

Lesson 7: Linear inequality word problems: foundations

Linear inequality word problems | Lesson

Google Classroom

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.

Statement	Inequality	Possible solutions
"It'll take at least $30$ ‍ minutes to get downtown."	$x \geq 30$ ‍	$30$ ‍ minutes, $45$ ‍ minutes, etc.
"I wouldn't pay more than $$ 6$ ‍ for a sandwich!"	$x \leq 6$ ‍	$$ 4$ ‍, $$ 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

\geq

what people will eat), but you also don't want to overspend (money spent on snacks

\leq

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

Khan Academy video wrapper

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.

Phrase	Translates to...
"More than $c$ ‍", "greater than $c$ ‍", or "higher than $c$ ‍"	$> c$ ‍
"Less than $c$ ‍" or "lower than $c$ ‍"	$< c$ ‍
"Greater than or equal to $c$ ‍" or "at least $c$ ‍"	$\geq c$ ‍
"Less than or equal to $c$ ‍" or "at most $c$ ‍"	$\leq c$ ‍
"No less than $c$ ‍"	$\geq c$ ‍
"No more than $c$ ‍"	$\leq c$ ‍
"Least", "lowest", or "minimum" value	The smallest value that satisfies the inequality
"Greatest", "highest", or "maximum" value	The largest value that satisfies the inequality
"A possible" value	Any 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

$ 0.40

and a pickle costs

$ 2.30

, what is greatest number of peppermint stick pickles Bryan can make if he has

$ 20

to buy the ingredients?

Each peppermint stick pickle requires a peppermint stick (

$ 0.40

) and a pickle (

$ 2.30

), which means the price of one peppermint stick pickle is:

$ 0.40 + $ 2.30 = $ 2.70

Let's use

x

to represent the number of peppermint stick pickles Bryan makes. The cost of

x

peppermint stick pickles is

2.7 x

dollars.

Bryan has

$ 20

, which means the cost of the ingredients he can afford must be less than or equal to

20

dollars.

Putting these two parts together gives us:

2.7 x \leq 20

We can find

x

by dividing both sides of the inequality by

2.7

\begin{aligned} 2.7 x & \leq 20 \\ \frac{2.7 x}{2.7} & \leq \frac{20}{2.7} \\ x & \leq 7.4 \end{aligned}

Note: even though

x

is less than or equal to a decimal, it's common sense that Bryan can't buy part of a peppermint stick or part of a pickle. Therefore, the answer to the question must be a whole number that satisfies the inequality.

Greatest means we're looking for the largest whole number that satisfies the inequality, which is

7

$7$ ‍ is the greatest number of peppermint stick pickles Bryan can make.

Try it!

Try: solve a linear inequality word problem

Zoey has

$ 5

and wants to rent a scooter. The scooter costs

$ 1

to unlock and

$ 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.

total cost =

Since Zoey has

$ 5

, the cost of renting the scooter must be

$ 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

Khan Academy video wrapper

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

$ 5,000

standard model and a

$ 7,000

racing model. Last month, his goal was to sell at least

36

scooters. If Diego met his goal and brought in over

$ 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?

Diego sold

s

standard model scooters and

r

racing model scooters last month. Since he met his goal of selling at least

36

scooters, the sum of

s

and

r

must be greater than or equal to

36

s + r \geq 36

Each standard model scooter sells for

$ 5,000

, so if Diego sells

1

scooter, he'll make

$ 5,000

, and if he sells

2

scooters, he'll make

$ 10,000

. In other words, Diego brought in

5,000 s

dollars from selling

s

standard model scooters.

Similarly, he brought in

7,000 r

dollars from selling

r

racing model scooters. Since he brought in over

$ 250,000

in sales, the sum of

5,000 s

and

7,000 r

must be greater than

250,000

5,000 s + 7,000 r > 250,000

Together, the system of inequalities is:

\begin{aligned} s + r \geq 36 \\ 5,000 s + 7,000 r > 250,000 \end{aligned}

Without additional constraints, there are many possible solutions for the system.

Let's check whether

s = 10

and

r = 30

satisfy both inequalities in the system:

\begin{aligned} s + r & \overset{?}{\geq} 36 \\ 10 + 30 & \overset{?}{\geq} 36 \\ 40 & \overset{✓}{\geq} 36 & satisfies first inequality \\ 5,000 s + 7,000 r & \overset{?}{>} 250,000 \\ 5,000 (10) + 7,000 (30) & \overset{?}{>} 250,000 \\ 50,000 + 210,000 & \overset{?}{>} 250,000 \\ 260,000 & \overset{✓}{>} 250,000 & satisfies second inequality \end{aligned}

Because

s = 10

and

r = 30

satisfy both inequalities,

(s, r) = (10, 30)

is a possible solution to the system.

Diego could have sold $10$ ‍ standard model scooters and $30$ ‍ racing model scooters.

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

$ 2

and

$ 5

each respectively. Her budget for the prizes can be no more than

$ 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?

Eugenia wants to buy at least

15

small prizes, which means

x

, the number of small prizes, must be greater than or equal to

15

x \geq 15

She also wants to buy at least

5

large prizes, which means

y

, the number of large prizes, must be greater than or equal to

5

y \geq 5

She wants at least

30

prizes total, which means the sum of

x

and

y

must be greater than or equal to

30

x + y \geq 30

Since each small prize costs

$ 2

x

small prizes cost

2 x

dollars. Similarly, since each large prize costs

$ 5

y

large prizes cost

5 y

dollars. Eugenia can spend at most

$ 100

on the prizes, which means the sum of

2 x

and

5 y

must be less than or equal to

100

2 x + 5 y \leq 100

Together, we have a system of four inequalities!

\begin{aligned} 2 x + 5 y \leq 100 \\ x + y \geq 30 \\ x \geq 15 \\ y \geq 5 \end{aligned}

If Eugenia buys

10

large prizes, what is a possible number of small prizes she can buy to satisfy the conditions described?

We're given the value of

y

, which means we need to find possible values of

x

that satisfy the system of inequalities in conjunction with the given

y

value.

We can substitute

10

into the first inequality for

y

and solve for

x

\begin{aligned} 2 x + 5 y & \leq 100 \\ 2 x + 5 (10) & \leq 100 \\ 2 x + 50 & \leq 100 \\ 2 x & \leq 50 \\ x & \leq 25 \end{aligned}

The first inequality tells us that Eugenia can buy at most

25

small prizes if she buys

10

large prizes. However, we also need to figure out the least number of small prizes she can buy to satisfy all of the conditions. Let's look at the second inequality:

\begin{aligned} x + y & \geq 30 \\ x + 10 & \geq 30 \\ x & \geq 20 \end{aligned}

The second inequality tells us that she must buy at least

20

small prizes if she wants to have at least

30

prizes total.

A quick look at the next two inequalities tells us that

20 \leq x \leq 25

satisfies the condition

x \geq 15

, and

y = 10

satisfies the condition

y \geq 5

If Eugenia buys $10$ ‍ large prizes, she can buy between $20$ ‍ and $25$ ‍ small prizes.

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.

Things to remember

Phrase	Translates to...
"More than $c$ ‍", "greater than $c$ ‍", or "higher than $c$ ‍"	$> c$ ‍
"Less than $c$ ‍" or "lower than $c$ ‍"	$< c$ ‍
"Greater than or equal to $c$ ‍" or "at least $c$ ‍"	$\geq c$ ‍
"Less than or equal to $c$ ‍" or "at most $c$ ‍"	$\leq c$ ‍
"No less than $c$ ‍"	$\geq c$ ‍
"No more than $c$ ‍"	$\leq c$ ‍
"Least", "lowest", or "minimum" value	The smallest value that satisfies the inequality
"Greatest", "highest", or "maximum" value	The largest value that satisfies the inequality
"A possible" value	Any value that satisfies the inequality

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peppermint stick pickles is crazy
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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.
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  just don't look at these charts. Imagine you are solving real problems. Then you will find it easy
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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
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  definitely, both methods work for any system of equations
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how did he take 10 and 30 in scooter question?
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- Kardokh Ashtab
  Posted 7 months ago. Direct link to Kardokh Ashtab's post “many solutions to the que...”
  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
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Debbiepearl
Posted 10 months ago. Direct link to Debbiepearl's post “The number of stops kayla...”
The number of stops kayla can take in the first question should be 8 and not 7.6 as stated in the video ,
3:18
of this lesson, because when we approximate 7.6 to the nearest whole number, it becomes 8;that is, it is rounded up.
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  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.
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is hit and trial the only way to solve system of linear equalities like shown in the example 1 ( diego sold...)
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How do you solve this? am confused

5,000s + 7000r > 250,000
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- StudyBuddy
  Posted 9 months ago. Direct link to StudyBuddy's post “Ok, so what are you tryin...”
  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?
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Guys please any help how to prepare? i have test on 9th of march
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Answer
- Ezra
  Posted 2 months ago. Direct link to Ezra's post “I've got a test on the 9t...”
  I've got a test on the 9th too. All I could advise is to finish the Khan Math Course, perfect your weaknesses, and take practice tests on Bluebook.
  Comment on Ezra's post “I've got a test on the 9t...”
  (4 votes)

Digital SAT Math

Course: Digital SAT Math > Unit 2

What are linear inequality word problems?

How do I write linear inequalities based on word problems?

Using inequalities to solve problems

Linear inequality word problems

What are some key phrases to look out for?

Let's look at some examples!

Try it!

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

Translating systems of inequalities word problems

Systems of linear inequalities word problems

Let's look at some examples!

Try it!

Your turn!

Things to remember

Want to join the conversation?