88 Chapter 2 Linear Equations and Inequalities in One Variable Objective 1 Examples For each inequality, graph the solution set and write the solution set in interval Objective 2 ▶ Solve a linear inequality using the addition and multiplication properties of inequalities. notation and set-builder notation. Inequality Graph Interval Notation Set-Builder Notation 1a. x > 4 –1 0 1 2 3 4 5 6 7 8 9 (4, ∞) 5xux > 46 1b. x≥-2 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 -2, ∞) 5xux≥-26 1c. 1 2 > x –5 –4 –3 –2 –1 0 1 2 3 4 5 ¢-∞, 1 2 ≤ bx` x < 1 2 r 1d. x ≤ 0 –5 –4 –3 –2 –1 0 1 2 3 4 5 (-∞, 0 5xux ≤ 06 1e. -1 < x < 3 –4 –3 –2 –1 0 1 2 3 4 5 6 (-1, 3) 5xu-1 < x < 36 Student Check 1 For each inequality, graph the solution set and write the solution set in interval notation and set-builder notation. a. x>-1 b. x ≥ 3 c. x<-1 d. x≤-3 e. 2 < x < 7 Note that Example 1 part (c) shows that the inequality 1 2 > x is equivalent to x < 1 2 . Example 1 parts (a)–(d) are illustrations of simple inequalities. The solutions of a simple inequality have to satisfy only one inequality. Example 1 part (e) is an illustration of a compound inequality. The solutions of a compound inequality must satisfy two inequalities, not just one. For instance, -1 < x < 3 means that “-1 < x and x < 3” or “x > -1 and x < 3” Compound inequalities will be studied more in Section 2.5. Addition and Multiplication Properties of Inequalities Solving linear inequalities is very similar to solving linear equations. The goal is the same, to isolate the variable on 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, they work with all inequality symbols. Property: Addition Property of Inequality If a, b, and c are real numbers, then a < b and a + c < b + c and a - c < b - c are equivalent inequalities. The property states that adding the same number to each side of an inequality or subtracting the same number from each side 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
hendricks_intermediate_algebra_1e_ch1_3
To see the actual publication please follow the link above