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Digital SAT Math
Course: Digital SAT Math > Unit 2
Lesson 7: Linear inequality word problems: foundationsLinear 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.
| Statement | Inequality | Possible solutions |
|---|---|---|
| "It'll take at least 30 minutes to get downtown." | x, is greater than or equal to, 30 | 30 minutes, 45 minutes, etc. |
| "I wouldn't pay more than dollar sign, 6 for a sandwich!" | x, is less than or equal to, 6 | dollar 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
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" | 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" 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, 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!
How do I write systems of linear inequalities based on word problems?
Translating systems of inequalities word problems
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!
Your turn!
Things to remember
| Phrase | Translates 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" 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 |
Want to join the conversation?
Isn't there 3 sided Inequalities like 1<2<4?(2 votes)
yeah, there is.(6 votes)
- why can't we just solve it like the system of linear equations? i mean solving the variables.(3 votes)
Perhaps won't yield a correct answer(3 votes)
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.(3 votes)- just don't look at these charts. Imagine you are solving real problems. Then you will find it easy(2 votes)
- How can you sovle two systems of inequalities with elimination?(1 vote)
- Convert it into equation first
But do note one thing if you divide both sides by a negative number inequality is gonna reverse.(3 votes)
i found it confusing.. i need tips to solve them.(2 votes)
The hardest part is the inequality symbols. Fix that and it’ll work out.(1 vote)
- is hit and trial the only way to solve system of linear equalities like shown in the example 1 ( diego sold...)(2 votes)
- I can't understand the Deigo's motorcycle problem?
why do we take 10 & 30?
we can also assume 15 & 25?(1 vote) 
I see there's some glitch in first question as I've tried every possible answer but it still is regarding it as wrong.(0 votes)- I believe there isn't, or at least I didn't have that problem(2 votes)