In this lesson, we will learn the difference between a conjunction and a disjunction and how to solve and graph ‘and’ and ‘or’ inequalities on the number line.

## What Are Inequalities?

Sydney has an important math test coming up on Friday. She wants to study for her test for at least 30 minutes. Sydney’s parents told her she needs to leave for swimming practice in 60 minutes. Therefore, she can study for no less than 30 minutes and no more than 60 minutes. This scenario can be written as a **compound inequality**. But what exactly does this mean?

You can think of an **inequality** as an equation, except that the equals sign is replaced with a less than or greater than sign. We still need to solve the inequality just like you would an equation. The only difference is that instead of one answer that makes the equation true, like *x* = 3, there are many answers that make an inequality true, like *x* ; 5. In this case, all numbers less than five would make the inequality true.

- 2
*x*+ 5 = 7 is an equation because it has an equals sign. - 2
*x*+ 5 ; 7 is an inequality because it has an inequality sign.

A **compound inequality** is just more than one inequality that we want to solve at the same time. We can either use the word ‘and’ or ‘or’ to indicate if we are looking at the solution to both inequalities (and), or if we are looking at the solution to either one of the inequalities (or).

*x* ; 7 and *x* ; -3, which can also be written as -3 ; *x* ; 7, is a compound inequality because it is two inequalities connected by the word ‘**and**‘. This is also known as a **conjunction**. In this case, we are looking for the solution to both inequalities. In other words, this solution satisfies both inequalities.

*x* ; 7 or *x* ; -3 is a compound inequality, also known as a **disjunction**, because it is two inequalities connected by the word ‘**or**‘. In this case, we are looking for a solution to either one of the equations.

Let’s check back in with Sydney. We know she needs to study for at least 30 minutes, but less than 60. If we set this up as a compound inequality, it looks like this: *x* > 30 and *x* < 60, also written as 30 < *x* < 60.

## How to Solve a Compound Inequality

#### Example 1

Let’s take a look at the inequality 2 + *x* ; 5 and -1 ; 2 + *x*, which can also be written as -1 ; 2 + *x* ; 5. This is a compound inequality because it uses the word ‘and.’ Now let’s go ahead and solve it.

1) Solve each part of the inequality separately.

2 + *x* < 5 and -1 < 2 + *x*

In the first equation, 2 + *x* < 5, we need to subtract 2 from each side to get the variable by itself. We then get *x* < 3.

In the second equation, -1 < 2 + *x*, we again subtract 2 from both sides. This gives us -3 < *x*.

Our solution, then, is *x* < 3 and -3 < x, or -3 < *x* < 3.

2) Graph on the number line.

Since this is a conjunction, the space between -3 and 3 is where the answer lies. In other words, any value between -3 and 3 satisfies this compound inequality.

Remember Sydney? If we were to display her inequality on a number line, it would show that all numbers between 30 and 60 would be possible solutions. Meaning, she could study for 35, minutes, 42 minutes and so on.

#### Example 2

But what if we are solving a disjunction? Let’s take a look at the following inequality: 7 ; 2*x* + 5 or 7 ; 5*x* – 3. This time the word ‘or’ is used instead of the word ‘and’. How do we solve this?

1) Solve each inequality:

For 7 ; 2*x* + 5, we subtract 5 from each side to get 2 ; 2*x*. Divide each side by 2 and we get 1 ; *x*.

For 7 ; 5*x* – 3, we add 3 to each side and get 10 ; 5*x*. Divide each side by 5 and we have 2 ; *x*.

2) Graph on the number line.

Since this is a disjunction, any value greater than 2 and less than 1 is where the answer lies. All real numbers satisfy this compound inequality.

## Important Notes

There are a few important points to keep in mind as you solve compound inequalities.

1.) If you want to check your answer or you are unsure of your answer, choose a value within the shaded region on the number line and plug it into both inequalities. If you get a true statement in both inequalities, you know the answer is correct. For example, when we solved 2 + *x* ; 5 and -1 ; 2 + *x* we found the solution was -3 ; *x* ; 3. 1 is a value between -3 and 3. Let’s plug it into the inequality to check our work.

- If we substitute 1 for
*x*in 2 +*x*< 5, we get 2 + 1 < 5, or 3 < 5. - If we substitute 1 for
*x*in -1 < 2 + x, we get -1 < 2 + 1, or -1 < 3.

Both of these are true statements, therefore our answer is correct.

2.) When graphing inequalities, use an open circle for less than or greater than and a shaded circle for less than or equal to and greater than or equal to. This tells us if the number shown on the number line is included (if it is shaded) or excluded (if it is open) as a solution to the inequality.

3.) If the arrows both point the same way, make sure to indicate where the compound inequality is true. For example, if you solved the inequality and got *x* > -3 or *x* > 5, it would look like this:

Since this is a disjunction, the shaded region greater than -3 satisfies this inequality. If it were a conjunction using ‘and,’ only the shaded region greater than 5 would satisfy the inequality.

## Lesson Summary

In this lesson, we learned that an **inequality** is like an equation except that it uses an inequality sign instead of an equals sign. We learned that a **conjunction** is two inequalities that use the word ‘**and**‘ and that the solution satisfies both inequalities. We also learned that a **disjunction** is a combination of two inequalities that uses the word ‘**or**‘ and that the solution satisfies either one of the equations.

To solve an inequality, first solve each inequality separately just like you would solve an equation. Once you have solved each part, graph the inequalities on the same number line. If it is a **conjunction** that uses the word and, the solution must work in both inequalities and the solution is in the overlap region of the graph. If it is a **disjunction** that uses the word or, the solution must work in either one of the equations. Always choose a value and plug it back into the original inequality in order to determine if the answer is correct.

## ‘And’ & ‘Or’ Compound Inequalities: Vocabulary

Vocabulary | Definitions |
---|---|

Inequality | like an equation where the equals sign is replaced with an inequality sign |

Compound inequality | more than one inequality that needs to be solved at the same time |

Conjunction | a compound inequality in which the solutions must work in both inequalities; connected by the word ‘and’ |

Disjunction | a compound inequality in which the solution must work in either one of the inequalities; uses the word ‘or’ |

## Learning Outcomes

After students finish this lesson, they should be able to:

- Define inequality and compound inequality
- Contrast conjunction and disjunction inequalities
- Solve compound inequalities and graph their solutions