### Learning Outcome

- Solve compound inequalities in the form of
*or*and express the solution graphically and with interval notation

## Solve Compound Inequalities in the Form of “*or”*

As we saw in the last section, the solution of a compound inequality that consists of two inequalities joined with the word *or* is the union of the solutions of each inequality. Unions allow us to create a new set from two that may or may not have elements in common.

In this section, you will see that some inequalities need to be simplified before their solution can be written or graphed.

In the following example, you will see an example of how to solve a one-step inequality in the *or* form. Note how each inequality is treated independently until the end where the solution is described in terms of both inequalities. You will use the same properties to solve compound inequalities that you used to solve regular inequalities.

### Example

Solve for *x*. [latex]3x–1<8[/latex] *or* [latex]x–5>0[/latex]

Remember to apply the properties of inequalities when you are solving compound inequalities. The next example involves dividing by a negative to isolate a variable.

### Example

Solve for *y*. [latex]2y+7\lt13\textit{ or }−3y–2\lt10[/latex]

In the last example, the final answer included solutions whose intervals overlapped. This caused the answer to include all the numbers on the number line. In words, we call this solution “all real numbers.” Any real number will produce a true statement for either [latex]y<3\text{ or }y\gt -4[/latex] when it is substituted for *y*.

### Example

Solve for [latex]z[/latex].

[latex]5z–3\gt−18[/latex] or [latex]−2z–1\gt15[/latex]

The following video contains an example of solving a compound inequality involving *or *and drawing the associated graph.

In the next section you will see examples of how to solve compound inequalities containing *and*.