Problem 65 - College Algebra Essentials

Question

Solve each absolute value inequality. $$|2(x-1)+4| \leq 8$$

Step-by-Step Solution

Answer

The solution to the inequality \(|2(x-1)+4| \leq 8\) is \(-1 \leq x \leq 3\).

Step 1: Breaking Up the Absolute Value Inequality

The absolute value in an inequality signifies that the expression within the absolute value notations can be either negative or positive. So, the given inequality \(|2(x-1)+4| \leq 8\) is equivalent to two separate inequalities: \(2(x-1)+4 \leq 8\) and \(-(2(x-1)+4) \leq 8\).

Step 2: Solving the First Inequality

Let's solve the first inequality \(2(x-1)+4 \leq 8\). Begin by distributing the 2 inside the parentheses to get \(2x - 2 + 4 \leq 8\). Simplify the left side to get \(2x + 2 \leq 8\). Next, subtract 2 from both sides to isolate the term with x: \(2x \leq 6\). Finally, divide each side by 2 to get \(x \leq 3\).

Step 3: Solving the Second Inequality

Now, let's solve the second inequality \(-(2(x-1)+4) \leq 8\). Begin by distributing the -1 inside the parentheses to get \(-2x + 2 + 4 \leq 8\). Simplify the left side to get \(-2x + 6 \leq 8\). Next, subtract 6 from both sides to isolate the term with x: \(-2x \leq 2\). Finally, divide each side by -2 (remember that when multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality changes) to get \(x \geq -1\).

Step 4: Determining the Solution Set

From the solutions of both inequalities, \(x \leq 3\) and \(x \geq -1\), the overlapping or common range of values for x is -1 up to 3. Thus, the solution set of the given absolute value inequality is \(-1 \leq x \leq 3\). This means any value of x within this range, inclusive of -1 and 3, will satisfy the inequality \(|2(x-1)+4| \leq 8\).