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# Solving Multi-Step Inequalities

Objective Learn to solve linear inequalities involving more than one operation.

This lesson is an extension of the methods you should already know to solve linear equations. The methods used are the Addition and Multiplication Properties of Inequalities, which parallel the corresponding properties for equalities. There are two significant distinctions that must be remembered.

The first is that when multiplying or dividing an inequality by a negative number, the direction of the inequality symbol must be changed.

The other is that the term “solve” is used a bit differently when speaking of inequalities. In equations, “solve” means to find a particular number that is the only solution to the equation. However, the result of solving an inequality is never a single number but rather the description of a set, such as x 5 or 22 x 7.

This distinction is very important.

## Solving Inequalities

Let's begin by reviewing the properties of inequalities.

Addition and Subtraction Properties of Inequalities

Adding or subtracting a fixed number to each side of an inequality produces an equivalent inequality. Any solution of either inequality is a solution of the other.

 x - 2 1 x - 2 + 2 1 + 2 Add 2 to each side. x 23

Adding 2 to each side of the original inequality shows that x - 2 1 is equivalent to x 3.

Multiplication and Division Properties of Inequalities

• Multiplying or dividing each side of an inequality by the same positive number produces an equivalent inequality.

• Multiplying or dividing by the same negative number produces an equivalent inequality if the direction of the inequality symbol is reversed.

 3x 12 -x 5 x 4 Divide each side by 3. x -5 Multiply each side by -1. So, 3x 12 is equivalent to x 4. So, -x 5 is equivalent to x -5.

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