# Math Models

## Solve Applications with Linear Inequalities


### Solve Applications with Linear Inequalities

Many real-life situations require us to solve inequalities. In fact, inequality applications are so common that we often do not even realize we are doing algebra. For example, how many gallons of gas can be put in the car for $20? Is the rent on an apartment affordable? Is there enough time before class to go get lunch, eat it, and return? How much money should each family member’s holiday gift cost without going over budget?
The method we will use to solve applications with linear inequalities is very much like the one we used when we solved applications with equations. We will read the problem and make sure all the words are understood. Next, we will identify what we are looking for and assign a variable to represent it. We will restate the problem in one sentence to make it easy to translate into an inequality. Then, we will solve the inequality.
Sometimes an application requires the solution to be a whole number, but the algebraic solution to the inequality is not a whole number. In that case, we must round the algebraic solution to a whole number. The context of the application will determine whether we round up or down. To check applications like this, we will round our answer to a number that is easy to compute with and make sure that number makes the inequality true.
A common goal of most businesses is to make a profit. Profit is the money that remains when the expenses have been subtracted from the money earned. In the next example, we will find the number of jobs a small businessman needs to do every month in order to make a certain amount of profit.
Sometimes life gets complicated! There are many situations in which several quantities contribute to the total expense. We must make sure to account for all the individual expenses when we solve problems like this.

### Key Concepts

1. Solving inequalities



### Section Exercises


### Practice Makes Perfect

Solve Applications with Linear Inequalities
In the following exercises, solve.

### Everyday Math


### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

### Chapter 3 Review Exercises


### 3.1 Using a Problem Solving Strategy

Approach Word Problems with a Positive Attitude
In the following exercises, reflect on your approach to word problems.
Use a Problem-Solving Strategy for Word Problems
In the following exercises, solve using the problem-solving strategy for word problems. Remember to write a complete sentence to answer each question.
Solve Number Problems
In the following exercises, solve each number word problem.

### 3.2 Solve Percent Applications

Translate and Solve Basic Percent Equations
In the following exercises, translate and solve.
Solve Percent Applications
In the following exercises, solve.
Find Percent Increase and Percent Decrease
In the following exercises, solve.
Solve Simple Interest Applications
In the following exercises, solve.
Solve Applications with Discount or Mark-up
In the following exercises, find the sale price.
In the following exercises, find ⓐ the amount of discount and ⓑ the sale price.
In the following exercises, find ⓐ the amount of discount and ⓑ the discount rate. (Round to the nearest tenth of a percent, if needed.)
In the following exercises, find ⓐ the amount of the mark-up and ⓑ the list price.

### 3.3 Solve Mixture Applications

Solve Coin Word Problems
In the following exercises, solve each coin word problem.
Solve Ticket and Stamp Word Problems
In the following exercises, solve each ticket or stamp word problem.
Solve Mixture Word Problems
In the following exercises, solve each mixture word problem.

### 3.4 Solve Geometry Applications: Triangles, Rectangles and the Pythagorean Theorem

Solve Applications Using Triangle Properties
In the following exercises, solve using triangle properties.
Use the Pythagorean Theorem
In the following exercises, use the Pythagorean Theorem to find the length of the hypotenuse.
In the following exercises, use the Pythagorean Theorem to find the length of the missing side. Round to the nearest tenth, if necessary.
In the following exercises, solve. Approximate to the nearest tenth, if necessary.
Solve Applications Using Rectangle Properties
In the following exercises, solve using rectangle properties.

### 3.5 Solve Uniform Motion Applications

Solve Uniform Motion Applications
In the following exercises, solve.

### 3.6 Solve Applications with Linear Inequalities

Solve Applications with Linear Inequalities
In the following exercises, solve.

### Practice Test

In the following exercises, use the Pythagorean Theorem to find the length of the missing side. Round to the nearest tenth, if necessary.