Solve each absolute value inequality. |3(x - 1)/4| < 6

Verified step by step guidance

1. Step 1: Understand the absolute value inequality. The inequality |3(x - 1)/4| < 6 means that the expression inside the absolute value, 3(x - 1)/4, must lie between -6 and 6. This can be rewritten as a compound inequality: -6 < 3(x - 1)/4 < 6.
2. Step 2: Eliminate the fraction by multiplying all parts of the inequality by 4 (the denominator). This gives: -24 < 3(x - 1) < 24.
3. Step 3: Simplify the inequality by dividing all parts by 3 (the coefficient of the term). This results in: -8 < x - 1 < 8.
4. Step 4: Solve for x by adding 1 to all parts of the inequality. This gives: -7 < x < 9.
5. Step 5: Write the solution as an interval. The solution to the inequality is all values of x between -7 and 9, which can be expressed as (-7, 9) in interval notation.

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 expressions 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 solve for the variable. This involves understanding how to manipulate and solve inequalities correctly.

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 solution set. It is essential to consider the implications of the inequality sign and ensure that the solution is expressed in interval notation or as a compound inequality.