Problem 67 - College Algebra Essentials

Question

Solve each absolute value inequality. $$\left|\frac{2 x+6}{3}\right|<2$$

Step-by-Step Solution

Answer

The solution to the absolute value inequality is \(x\) is in the interval \((-6,0)\)

Step 1: Understand the absolute value properties

Going by the properties of absolute value, an expression \(|a| < b\) is equivalent to \( -b < a < b \). So, we can rewrite \( \left|\frac{2 x+6}{3}\right|<2 \) as \( -2 < \frac{2 x+6}{3} < 2 \)

Step 2: Solve the inequality

Now solve the inequality for \(x\). First, multiply all parts by 3 to remove the denominator: \( -6 < 2x+6 < 6 \). Next, subtract 6 from all parts to isolate \(x\): \( -12 < 2x < 0 \). Lastly, divide all parts by 2 to solve for \(x\): \( -6 < x < 0 \)

Step 3: Write the solution in interval notation

The solution set for the inequality \( -6 < x < 0 \) written in interval notation is \((-6,0)\). In interval notation, parentheses indicate that the endpoints are not included.