Problem 68 - College Algebra Essentials

Question

Solve each absolute value inequality. $$\left|\frac{3(x-1)}{4}\right|<6$$

Step-by-Step Solution

Answer

The solution to \( \left|\frac{3(x-1)}{4}\right| < 6 \) is \( x \) in the interval \( (-7, 9) \).

Step 1: Interpret the Absolute Value Inequality

The absolute value of a number is its distance from zero. Therefore, \( \left|\frac{3(x-1)}{4}\right| < 6 \) means the expression \( \frac{3(x-1)}{4} \) is less than 6 units from 0 on the number line, resulting in a compound inequality which can be written as \( -6 < \frac{3(x-1)}{4} < 6 \).

Step 2: Simplify Compound Inequalities

Now simplify the two inequalities separately. To do that you need to multiply every part of the inequality by 4, which gives \( -24 < 3(x - 1) < 24 \). Now, distribute 3 on the left side of the compound inequality: \( -24 < 3x - 3 < 24 \). Then, to isolate 'x', add 3 to all parts of the inequality, resulting in \( -21 < 3x < 27 \). Finally, divide each part of the inequality by 3 to get the final result: \( -7 < x < 9 \).

Step 3: Writing the Answer in Interval Notation

The solution of the inequality can also be written in interval notation. The interval notation for \( -7 < x < 9 \) is \( (-7, 9) \).