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**4.1 Solving Linear Inequalities**

Objective: Solve and graph simple and compound inequalities in one variable.

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What are inequalities?

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**Inequality Symbols Greater than: > Less than: <**

Greater than or equal to: ≥ Less than or equal to: ≤

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Inequalities Inequalities such as x ≤ 1 and p – 3 > 7 are linear inequalities in one variable. A solution of an inequality in one variable is a value of the variable that makes the inequality true. Example: -4, 0.7, and 1 are solutions of x ≤ 1 Two inequalities are equivalent if they have the same solutions

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**Properties of Inequalities**

To write an equivalent inequality: Add the same number to each side. Subtract the same number from each side. Multiply or divide each side by the same positive number. Multiply or divide each side by the same negative number and reverse the inequality symbol.

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Solve the inequality. x – 4 > -6 -5y + 2 ≥ -13

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Solve the inequality. 7 – 4x < 1 – 2x 2x – 3 > x

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Solve -x + 3 ≤ -6 3y – 5 < 10 x + 3 < 8 2x – 3 > x

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**Graphing Inequalities**

The graph of an inequality in one variable consists of all points on a real number line that are solutions of the inequality. To graph an inequality in one variable: Use an open dot () for < or > Use a solid dot () for ≤ or ≥

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**Graphing Inequalities**

Graph x < 2 Graph x ≥ 1

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**Solve the inequality and then graph your solution.**

4x + 3 ≤ 6x – 5 -x + 2 < -3

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**Compound Inequalities**

A compound inequality is two simple inequalities joined by the word “and” or the word “or” AND All real numbers greater than or equal to -2 and less than 1 can be written as: -2 ≤ x < 1 Graph: OR All real numbers less than -1 or greater than or equal to 2 can be written as: x < -1 or x ≥ 2 Graph:

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**Graph the Compound Inequality**

x < -2 or x ≥ 3 x ≤ -2 or x > 3 -2 < x ≤ 3

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