Problem 81 - College Algebra Essentials

Question

Solve each absolute value inequality. $$-2|x-4| \geq-4$$

Step-by-Step Solution

Answer

The solution to the inequality is \(2 \leq x \leq 6\).

Step 1: Division

Divide both sides of the inequality by -2. Remember that dividing or multiplying an inequality by a negative number switches the direction of the inequality sign. This gives \(|x-4| \leq 2\).

Step 2: Split the Absolute Value Inequality

Split the absolute value inequality into two separate inequalities: \(x-4 \leq 2\) and \(-(x-4) \leq 2\).

Step 3: Solve each Inequality

Solving the two inequalities will give: For \(x-4 \leq 2\), adding 4 to both sides gives \(x \leq 6\).For \(-(x-4) \leq 2\), distribute the negative, giving \(-x + 4 \leq 2\). Subtracting 4 from both sides gives \(-x \leq -2\). Multiplying each side by -1 (and flipping the inequality sign as per Step 1) gives \(x \geq 2\).

Step 4: Write down the solution

The solutions to the original inequality are values of x that satisfy \(both\) inequalities. That means \(x\) must be between 2 and 6, including the endpoints.