Section 2.5 Compound Inequalities 107 This three-part inequality can be solved by applying the same operation to each part. The goal is to isolate the variable in the middle. Subtract 3 from each part. Simplify. Divide each part by 4. Simplify. -5 ≤ 2x - 1 ≤ 5 -5 + 1 ≤ 2x - 1 + 1 ≤ 5 + 1 -4 ≤ 2x ≤ 6 -4 2 ≤ 2x 2 ≤ 6 2 -2 ≤ x ≤ 3 3e. -7 ≤ 4x + 3 ≤ 10 -7 - 3 ≤ 4x + 3 - 3 ≤ 10 - 3 -10 ≤ 4x ≤ 7 -10 4 ≤ 4x 4 ≤ 7 4 - 5 2 ≤ x ≤ 7 4 So, the graph of the solution set is – 7 5 2 4 –5 –4 –3 –2 –1 0 1 2 4 5 6 The solution set is c- 5 2 , 7 4 d . Student Check 3 Solve each compound inequality. Write each solution set in interval notation and graph the solution set. Solve (d) also using the compact form. a. x + 4 ≥ 1 and -3x≥-6 b. 2y - 3 > 7 and 4(y + 2) > 4 c. 8 + 5x>-2 and 7x≤-21 d. 1 - 9x < 8 and 1 - 9x>-8 e. -6 ≤ 7x + 5 ≤ 19 Compound Inequalities Joined by “Or” We will now focus on compound inequalities joined by the term “or.” Recall that a compound sentence joined by the term “or” is a statement in which one of two conditions must be met. Suppose the advertisement for a job states that “Candidates must have a Bachelor’s degree or candidates must have 5 years of experience.” In this case, only one of the conditions is necessary for a person to apply for the job. Note a person may also meet both conditions and be eligible to apply. Suppose we want to solve the inequality: x<-1or x > 5. Solutions of this compound inequality are numbers that satisfy at least one part of the inequality. Consider the possible solutions. Value x < -1 or x > 5 Is the value a solution? x = 3 3 < -1 or 3 > 5 Because 3 makes both inequalities false, the compound inequality is false. So, 3 is a not a solution of the compound inequality. x =- 2 -2 < -1 or -2 > 5 Because -2 makes the first inequality true, the compound inequality is true. So, -2 is a solution of the compound inequality. x = 7 7 < -1 or 7 > 5 Because 7 makes the second inequality true, the compound inequality is true. So, 7 is a solution of the compound inequality. Objective 4 ▶ Solve compound inequalities involving “or.” Add 1 to each part. Simplify. Divide each part by 2. Simplify.
hendricks_intermediate_algebra_1e_ch1_3
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