### Learning Objectives

- Solve multi-step inequalities
- Combine properties of inequality to isolate variables, solve algebraic inequalities, and express their solutions graphically
- Simplify and solve algebraic inequalities using the distributive property to clear parentheses and fractions

## Combine properties of inequality to solve algebraic inequalities

A popular strategy for solving equations, isolating the variable, also applies to solving inequalities. By adding, subtracting, multiplying and/or dividing, you can rewrite the inequality so that the variable is on one side and everything else is on the other. As with one-step inequalities, the solutions to multi-step inequalities can be graphed on a number line.

### Example

Solve for *p*. [latex]4p+5<29[/latex]

Check the solution.

### Example

Solve for *x*: [latex]3x–7\ge 41[/latex]

Check the solution.

When solving multi-step equations, pay attention to situations in which you multiply or divide by a negative number. In these cases, you must reverse the inequality sign.

### Example

Solve for *p*. [latex]−58>14−6p[/latex]

Check the solution.

In the following video, you will see an example of solving a linear inequality with the variable on the left side of the inequality, and an example of switching the direction of the inequality after dividing by a negative number.

In the following video, you will see an example of solving a linear inequality with the variable on the right side of the inequality, and an example of switching the direction of the inequality after dividing by a negative number.

## Simplify and solve algebraic inequalities using the distributive property

As with equations, the distributive property can be applied to simplify expressions that are part of an inequality. Once the parentheses have been cleared, solving the inequality will be straightforward.

### Example

Solve for *x*. [latex]2\left(3x–5\right)\leq 4x+6[/latex]

Check the solution.

In the following video, you are given an example of how to solve a multi-step inequality that requires using the distributive property.

### Think About It

In the next example, you are given an inequality with a term that looks complicated. If you pause and think about how to use the order of operations to solve the inequality, it will hopefully seem like a straightforward problem. Use the textbox to write down what you think is the best first step to take.

Solve for a. [latex] \displaystyle\frac{{2}{a}-{4}}{{6}}{<2}[/latex]

Check the solution.

## Summary

Inequalities can have a range of answers. The solutions are often graphed on a number line in order to visualize all of the solutions. Multi-step inequalities are solved using the same processes that work for solving equations with one exception. When you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality symbol. The inequality symbols stay the same whenever you add or subtract *either positive or negative* numbers to both sides of the inequality.