Page 129

hendricks_intermediate_algebra_1e_ch1_3

Section 2.7 Absolute Value Inequalities 127 Absolute Value Inequalities with > or ≥ The graph from Objective 1 will help us solve the equation uxu ≥ 2. x –5 –4 –3 –2 –1 0 1 2 3 4 5 |x| 5 4 3 2 1 0 1 2 3 4 5 The numbers larger than and including 2 or less than and including -2 have an absolute value greater than or equal to 2. So, the solution set is (-∞, -2 ∪ 2, ∞). This leads to a property that enables us to solve absolute value inequalities of the form uXu > k or uXu ≥ k. Property: Property 2 for Absolute Value Inequalities Let k be a positive real number. If uXu > k, then X<-k or X > k. If uXu ≥ k, then X≤-k or X ≥ k. Procedure: Solving an Inequality of the Form |X| > k or |X| ≥ k Step 1: Isolate the absolute value expression on one side of the inequality. Step 2: Apply property 2 to remove the absolute value sign. Step 3: Solve the resulting compound inequality. Step 4: Graph the solution set and write the answer in interval notation. Objective 2 Examples Solve each absolute value inequality. 2a. uxu > 5 2b. uy + 3u ≥ 4 2c. u2x - 1u - 5 > 4 Solutions 2a. uxu > 5 x > 5 or x < -5 Graph: –10 –8 –6 –4 –2 0 2 4 6 8 10 Interval: (-∞, -5) ∪ (5, ∞) 2b. uy + 3u ≥ 4 y + 3 ≥ 4 or y + 3 ≤ -4 y + 3 - 3 ≥ 4 -3 y + 3 - 3≤-4 - 3 y ≥1 y≤-7 Graph: –12 –10 –8 –6 –4 –2 0 2 4 6 8 Interval: (-∞, -7 ∪ 1, ∞) Objective 2 ▶ Solve absolute value inequalities involving > or ≥. Apply property 2. Apply property 2. Subtract 3 from each side. Simplify.


hendricks_intermediate_algebra_1e_ch1_3
To see the actual publication please follow the link above