Problem 80 - College Algebra Essentials

Question

Solve each absolute value inequality. $$5|2 x+1|-3 \geq 9$$

Step-by-Step Solution

Answer

The solutions to the absolute value inequality are \(x \geq 0.7\) and \(x \leq -1.7\).

Step 1: Isolate the Absolute Value

Add 3 on both the sides to isolate the absolute value, resulting in \(5|2x+1| \geq 12\). Now divide both sides by 5 to isolate the absolute value completely, which gives \(|2x+1| \geq 2.4 \)

Step 2: Case-wise Split

Split the inequality into two cases to remove the absolute value: Case 1: \(2x+1 \geq 2.4\) and Case 2: \(-(2x+1) \geq 2.4\)

Step 3: Solve the Inequalities

Solve these two inequalities separately. For case 1, subtract 1 from both sides giving: \(2x \geq 1.4\) and then divide by 2, producing \(x \geq 0.7\). For case 2, distribute the negative sign giving: \(-2x-1 \geq 2.4\). Re-arrange to get \(2x \leq -3.4\) and divide by 2 to yield \(x \leq -1.7\).

Step 4: Verify Against Original Inequation

Plug in \(x = 0.7\) and \(x = -1.7\) and verify that both satisfy the initial inequality. After verifying, report the solution.