In Exercises 59–94, solve each absolute value inequality. |3x + 5| < 17

Verified step by step guidance

1. Recall that an absolute value inequality of the form \(|A| < B\) (where \(B > 0\)) can be rewritten as a compound inequality: \(-B < A < B\).
2. Apply this rule to the given inequality \(|3x + 5| < 17\), which becomes \(-17 < 3x + 5 < 17\).
3. Next, solve the compound inequality by isolating \(x\). Start by subtracting 5 from all parts: \(-17 - 5 < 3x < 17 - 5\).
4. Simplify the inequalities: \(-22 < 3x < 12\).
5. Finally, divide all parts by 3 to solve for \(x\): \(\frac{-22}{3} < x < 4\).

Key Concepts

Absolute Value Definition

The absolute value of a number represents its distance from zero on the number line, always as a non-negative value. For any expression |A|, it equals A if A is non-negative, and -A if A is negative. Understanding this helps in rewriting absolute value inequalities into equivalent compound inequalities.

Solving Absolute Value Inequalities

An inequality of the form |A| < B (where B > 0) can be rewritten as a compound inequality: -B < A < B. This means the expression inside the absolute value lies between -B and B. This approach allows solving the inequality by breaking it into two linear inequalities.

Linear Inequalities

Once the absolute value inequality is rewritten without absolute values, it becomes a linear inequality or a system of inequalities. Solving these involves isolating the variable using algebraic operations like addition, subtraction, multiplication, or division, while considering the direction of the inequality.