In Exercises 59–94, solve each absolute value inequality. |3(x - 1)/4| < 6

Verified step by step guidance

1. Start by understanding the absolute value inequality $|\frac{3(x-1)}{4}|<6$. This means that the expression inside the absolute value, $\frac{3(x-1)}{4}$, must be between -6 and 6.
2. Rewrite the inequality without the absolute value as a compound inequality: $-6<\frac{3(x-1)}{4}<6$.
3. To eliminate the fraction, multiply all parts of the inequality by 4: $-6\times 4<3(x-1)<6\times 4$.
4. Simplify the inequality: $-24<3(x-1)<24$.
5. Divide each part of the inequality by 3 to solve for $x$: $-\frac{24}{3}<x-1<\frac{24}{3}$.

Key Concepts

Absolute Value

Absolute value measures the distance of a number from zero on the number line, regardless of direction. For any real number x, the absolute value is denoted as |x| and is defined as |x| = x if x ≥ 0, and |x| = -x if x < 0. Understanding absolute value is crucial for solving inequalities that involve expressions within absolute value symbols.

Inequalities

Inequalities express a relationship between two values that are not necessarily equal, using symbols such as <, >, ≤, or ≥. In the context of absolute value inequalities, we often need to split the inequality into two separate cases to find the solution set. This involves considering both the positive and negative scenarios of the expression inside the absolute value.

Solving Absolute Value Inequalities

To solve an absolute value inequality like |A| < B, we translate it into two separate inequalities: -B < A < B. This method allows us to isolate the variable and find the range of values that satisfy the original inequality. It is essential to correctly interpret the inequality signs and ensure that the solution is expressed in interval notation or as a compound inequality.