Problem 93 - College Algebra Essentials

Question

Solve each absolute value inequality. $$4+\left|3-\frac{x}{3}\right| \geq 9$$

Step-by-Step Solution

Answer

\(x \leq -6, x \geq 24\)

Step 1: Isolate the absolute value

First, isolate the absolute value on one side of the inequality. You can do this by subtracting 4 from both sides of the inequality, obtaining \(\left|3-\frac{x}{3}\right| \geq 5\).

Step 2: Consider the two cases

Now separate into two cases. This is because an absolute value effectively 'removes' a negative sign, so we must consider case 1 where \(3-\frac{x}{3} \geq 5\) (for positive values) and case 2 where \(-(3-\frac{x}{3}) \geq 5\) (for negative values).

Step 3: Solve the first inequality

In case 1, subtract 3 from both sides and multiply by 3 to isolate x, to get \(-\frac{x}{3} \geq 2\) or equivalently, \(x \leq -6\).

Step 4: Solve the second inequality

In case 2, distribute the negative sign to get \(\frac{x}{3} - 3 \geq 5\). Then, add 3 to both sides, and multiply by 3 to isolate x, to get \(\frac{x}{3} \geq 8\) or equivalently, \(x \geq 24\).

Step 5: Combining the solutions

Combining the solution from both the cases we get the final answer.