Problem 86 - College Algebra Essentials

Question

Solve each absolute value inequality. $$9 \leq|4 x+7|$$

Step-by-Step Solution

Answer

The solutions to the absolute value inequality \(9 \leq|4x + 7|\) are all \(x \leq -4\) or \(x \geq 0.5\).

Step 1: Isolate Absolute Value

The absolute value expression \(|4x + 7|\) is already isolated.

Step 2: Convert Absolute Value Inequality

Next, we convert the absolute value inequality into two separate inequalities by using the definition of absolute value. So, \(|4x + 7|\) becomes two inequalities.1. \(4x + 7 \geq 9\) when \(4x + 7\) is positive or zero.2. \(-(4x + 7) \geq 9\) when \(4x + 7\) is negative.

Step 3: Solve Resulting Inequalities

Now we'll solve the two inequalities we derived from the absolute value inequality:1. Solving \(4x + 7 \geq 9\), we subtract 7 from both sides, and then divide by 4, giving \(x \geq 0.5\)2. For \-(4x + 7) \geq 9\), which simplifies to \(-4x -7 \geq 9\), we add 7 to both sides then divide by -4, giving \(x \leq -4\).

Step 4: Combine Solutions

Since the given inequality is \(9 \leq|4x + 7|\), we need to consider the solutions that satisfy either \(x \geq 0.5\) or \(x \leq -4\). Hence, these solutions are all the values of \(x\) that satisfy the given inequality.