Solving Inequalities in One Variable Section 2.8 Solving Inequalities in One Variable

Inequalities An inequality is a statement that describes how two numbers are related to one another. Smaller numbers are on the left. 2 – 2 1 3 4 5 – 1 – 3 – 4 – 5 – 4 < – 1 – 1 > – 4 “is less than” “is greater than” 3 < 10 4 > – 3 – 2 < 4 2

Example In each statement, replace the question mark with the symbol. a. 7 ? –1 b. –4 ? 2 c. 0 ? –5 a. 7 > –1 b. –4 < 2 c. 0 > –5 Use >, since 7 is to the right of –1 on the number line. Use <, since –4 is to the left of 2 on the number line. Use >, since 0 is to the right of –5 on the number line.

Graphing an Inequality Any number that makes an inequality true is called a solution of the inequality. The set of all numbers that makes the inequality true is called the solution set. -5 -4 -3 -2 -1 1 2 3 4 5 x ≤ 3 A closed circle is used to show that the endpoint is included in the answer. The symbols  and  will use a closed circle. 4

Graphing an Inequality x < 5 -5 -4 -3 -2 -1 1 2 3 4 5 An open circle is used to show that the endpoint is NOT included in the answer. The symbols > and < will use an open circle. 5

Example State each mathematical relationship in words and then graph it. a. x < –4 b. x ≥ –3 c. –1 < x d. x is less than –4 x is greater than or equal to –3 x is greater than or equal to –1 x is less than or equal to –1/2

Example Translate each English statement into an algebraic statement. a. To achieve a passing grade in her class, Patty must get at least an 80% on her final. x ≥ 80 b. Fewer than 200 students were able to attend the assembly. x < 200 7

Procedure for Solving Inequalities You may use the same procedure to solve inequalities that you used to solve equations except that the direction of the inequality is reversed if you multiply or divide both sides by a negative number.

Example Solve and graph. 3x + 7 ≥ 13 3x + 7 – 7 ≥ 13 – 7 3x ≥ 6 x ≥ 2

Example Solve and graph. 5 – 3x > 7 –3x > 2

Example Solve and graph. Continued

Example (cont)