Solve each equation or inequality. | 5x + 2 | - 2 < 3

Verified step by step guidance

1. Start by isolating the absolute value expression on one side of the inequality. Add 2 to both sides to get: \(| 5x + 2 | < 3 + 2\).
2. Simplify the right side to rewrite the inequality as: \(| 5x + 2 | < 5\).
3. Recall that for an inequality of the form \(|A| < B\), where \(B > 0\), the solution is \(-B < A < B\). Apply this to get: \(-5 < 5x + 2 < 5\).
4. Solve the compound inequality by subtracting 2 from all parts: \(-5 - 2 < 5x < 5 - 2\), which simplifies to \(-7 < 5x < 3\).
5. Finally, divide all parts of the inequality by 5 to isolate \(x\): \(\frac{-7}{5} < x < \frac{3}{5}\). This is the solution set for the inequality.

Key Concepts

Absolute Value Inequalities

Absolute value inequalities involve expressions where the absolute value of a variable or expression is compared to a number. To solve them, you consider the definition of absolute value as distance from zero, leading to two cases: one positive and one negative. For example, |A| < B means -B < A < B.

Isolating the Absolute Value Expression

Before solving an absolute value inequality, isolate the absolute value term on one side of the inequality. This often involves adding or subtracting constants and dividing or multiplying by coefficients. Proper isolation is crucial to correctly apply the definition of absolute value.

Solving Linear Inequalities

After removing the absolute value by splitting into two inequalities, solve each linear inequality separately. This involves standard algebraic techniques like adding, subtracting, multiplying, or dividing both sides by constants, while remembering to reverse inequality signs when multiplying or dividing by negative numbers.