# Solving Linear Equations

## Solve Compound Inequalities


### Solve Compound Inequalities with “and”

Now that we know how to solve linear inequalities, the next step is to look at compound inequalities. A compound inequality is made up of two inequalities connected by the word “and” or the word “or.” For example, the following are compound inequalities.
To solve a compound inequality means to find all values of the variable that make the compound inequality a true statement. We solve compound inequalities using the same techniques we used to solve linear inequalities. We solve each inequality separately and then consider the two solutions.
To solve a compound inequality with the word “and,” we look for all numbers that make both inequalities true. To solve a compound inequality with the word “or,” we look for all numbers that make either inequality true.
Let’s start with the compound inequalities with “and.” Our solution will be the numbers that are solutions to both inequalities known as the intersection of the two inequalities. Consider the intersection of two streets—the part where the streets overlap—belongs to both streets.
To find the solution of the compound inequality, we look at the graphs of each inequality and then find the numbers that belong to both graphs—where the graphs overlap.
For the compound inequality  and  we graph each inequality. We then look for where the graphs “overlap”. The numbers that are shaded on both graphs, will be shaded on the graph of the solution of the compound inequality. See .
We can see that the numbers between  and  are shaded on both of the first two graphs. They will then be shaded on the solution graph.
The number  is not shaded on the first graph and so since it is not shaded on both graphs, it is not included on the solution graph.
The number two is shaded on both the first and second graphs. Therefore, it is be shaded on the solution graph.
This is how we will show our solution in the next examples.
Sometimes we have a compound inequality that can be written more concisely. For example,  and  can be written simply as  and then we call it a double inequality. The two forms are equivalent.
To solve a double inequality we perform the same operation on all three “parts” of the double inequality with the goal of isolating the variable in the center.
When written as a double inequality,  it is easy to see that the solutions are the numbers caught between one and five, including one, but not five. We can then graph the solution immediately as we did above.
Another way to graph the solution of  is to graph both the solution of  and the solution of  We would then find the numbers that make both inequalities true as we did in previous examples.

### Solve Compound Inequalities with “or”

To solve a compound inequality with “or”, we start out just as we did with the compound inequalities with “and”—we solve the two inequalities. Then we find all the numbers that make either inequality true.
Just as the United States is the union of all of the 50 states, the solution will be the union of all the numbers that make either inequality true. To find the solution of the compound inequality, we look at the graphs of each inequality, find the numbers that belong to either graph and put all those numbers together.
To write the solution in interval notation, we will often use the union symbol,  to show the union of the solutions shown in the graphs.

### Solve Applications with Compound Inequalities

Situations in the real world also involve compound inequalities. We will use the same problem solving strategy that we used to solve linear equation and inequality applications.
Recall the problem solving strategies are to first read the problem and make sure all the words are understood. Then, identify what we are looking for and assign a variable to represent it. Next, restate the problem in one sentence to make it easy to translate into a compound inequality. Last, we will solve the compound inequality.

### Key Concepts

1. How to solve a compound inequality with “and”


2. Double Inequality


3. How to solve a compound inequality with “or”



### Practice Makes Perfect

Solve Compound Inequalities with “and”
In the following exercises, solve each inequality, graph the solution, and write the solution in interval notation.
Solve Compound Inequalities with “or”
In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.
Mixed practice
In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.
Solve Applications with Compound 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?