104 Chapter 2 Linear Equations and Inequalities in One Variable Student Check 2 Find the union of the sets. Draw the graph of each union and write the solution set in interval notation and set-builder notation. a. A = 4, ∞) and B = (-∞, 0 b. A = (-∞, -2) and B = (-∞, 3) c. A = -1, ∞) and B = (-∞, 5 Compound Inequalities Joined by “And” Compound inequalities are mathematical compound sentences. Recall that a compound sentence is formed by joining two simple sentences by a coordinating conjunction, such as “and” or “or.” A compound inequality is defined similarly. Definition: A compound inequality consists of two inequalities joined by the terms “and” or “or.” We will first focus on compound inequalities joined by the term “and.” Recall that a compound sentence joined by the term “and” is a statement in which two conditions must be met. Suppose an advertisement for a job states that “Candidates must have a Bachelor’s degree and candidates must have 5 years of experience.” Both conditions must be met for a person to be eligible to apply for the job. Suppose we want to solve the inequality: x≥-1and x ≤ 5. Solutions of this compound inequality are numbers that satisfy both parts of the inequality. Consider the possible solutions. Value x ≥ -1 and x ≤ 5 Is the value a solution? x = 3 3 ≥ -1 and 3 ≤ 5 Because 3 makes both inequalities true, the compound inequality is true. So, 3 is a solution of the compound inequality. x =- 2 -2 ≥ -1 and -2 ≤ 5 Because -2 makes only the second inequality true, the compound inequality is false. So, -2 is not a solution of the compound inequality. So, solutions of the compound inequality, x≥-1 and x ≤ 5, are values that are both greater than or equal to -1 and also less than or equal to 5. The solutions lie between and include -1and 5. This can be written in a more compact way as -1 ≤ x ≤ 5. Thus, the solution set is -1, 5 or 5x k-1 ≤ x ≤ 56. The values of the variable that make both inequalities true are solutions of a compound inequality joined by “and.” So, solutions must lie in the intersection of the solution sets of each individual inequality in the compound inequality. Procedure: Solving a Compound Inequality Involving “And” Step 1: Find the solution set of inequality 1. Step 2: Find the solution set of inequality 2. Step 3: Find the intersection of the solution sets of inequalities 1 and 2. Step 4: Write the final solution set in interval notation and provide its graph. Objective 3 Examples Solve each compound inequality. Write each solution set in interval notation and graph the solution set. 3a. x + 5 ≥ 8 and -2x≥-10 3b. 3y + 1 < 5 and 2(y - 3) < 2 3c. 4 - 2x > 0 and 5x ≥ 15 3d. 2x - 1 ≤ 5 and 2x - 1≥-5 3e. -7 ≤ 4x + 3 ≤ 10 Objective 3 ▶ Solve compound inequalities involving “and.”
hendricks_intermediate_algebra_1e_ch1_3
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