Section 2.7 Linear Inequalities in One Variable 169 Student Check 3 Graph each inequality and express the solution set in set-builder notation. a. x > -2 b. x ≤ -4.2 c. 5 3 > x d. -4 ≤ x < 2 Linear Inequalities Solving linear inequalities is very similar to solving linear equations. The goal is the same: to isolate the variable to one side of the inequality. The properties that enable us to do this are the addition and multiplication properties of inequality. While only one inequality symbol is used in the statement of the following properties, the properties work with all inequality symbols. Property: Addition Property of Inequality If a < b and c is a real number, then a + c < b + c and a - c < b - c. The property illustrates that adding or subtracting a number from both sides of an inequality produces an equivalent inequality. For an illustration of this property, let a = -5, b = 2, and c = 4. a < b a + c < b + c a - c < b - c -5 < 2 -5 + 4 < 2 + 4 -5 - 4 < 2 - 4 True -1 < 6 -9 < -2 True True Adding or subtracting a number from both sides of an inequality maintains the inequality relationship. Property: Multiplication Property of Inequality 1. If a < b and c > 0, then ac < bc and a c < b c . 2. If a < b and c < 0, then ac > bc and a c > b c . This property states the following: 1. Multiplying or dividing by a positive number produces an equivalent inequality. 2. Multiplying or dividing by a negative number produces an equivalent inequality only if the inequality symbol is reversed. For an illustration of the property, let a = -6 and b = 12. Let c = 3. a < b ac < bc a c < b c -6 < 12 -6(3) < 12(3) -6 3 < 12 3 True -18 < 36 -2 < 4 True True Objective 4 ▶ Solve linear inequalities.
hendricks_beginning_intermediate_algebra_1e_ch1_3
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