Solve each equation or inequality.

Verified step by step guidance

1. Recognize that the absolute value expression \(|18 - 3x|\) represents the distance of \$18 - 3x$ from zero on the number line, and absolute values are always greater than or equal to zero.
2. Note that the inequality is \(|18 - 3x| + 4 < -13\). Since \(|18 - 3x| \geq 0\), the smallest value of \(|18 - 3x| + 4\) is 4, which is already greater than -13.
3. Because the left side \(|18 - 3x| + 4\) can never be less than a negative number like -13, there are no values of \(x\) that satisfy this inequality.
4. Conclude that the inequality has no solution since an absolute value plus a positive number cannot be less than a negative number.
5. Therefore, the solution set is the empty set, often denoted as \(\emptyset\).

Key Concepts

Absolute Value Inequalities

Absolute value inequalities involve expressions where the absolute value of a variable or expression is compared to a number. Understanding how to interpret and solve inequalities like |A| < B or |A| > B is essential, where the solution depends on whether B is positive, zero, or negative.

Properties of Absolute Value

The absolute value of a number represents its distance from zero on the number line and is always non-negative. This means |x| ≥ 0 for any real x, and |x| < 0 has no solution. Recognizing this helps determine if an inequality involving absolute values is possible or not.

Solving Linear Inequalities

Solving linear inequalities involves isolating the variable and understanding inequality rules, such as reversing the inequality sign when multiplying or dividing by a negative number. This skill is necessary when breaking down absolute value inequalities into compound inequalities.