Solve each equation or inequality. | 3x + 1 | - 1 < 2

Verified step by step guidance

1. Start by isolating the absolute value expression on one side of the inequality. Add 1 to both sides to get: \(| 3x + 1 | < 3\).
2. Recall that the inequality \(|A| < B\) means that \(-B < A < B\). Apply this to the inequality: \(-3 < 3x + 1 < 3\).
3. Break the compound inequality into two separate inequalities: \(-3 < 3x + 1\) and \$3x + 1 < 3$.
4. Solve each inequality for \(x\). For \(-3 < 3x + 1\), subtract 1 from both sides to get \(-4 < 3x\), then divide by 3 to get \(\frac{-4}{3} < x\). For \$3x + 1 < 3\(, subtract 1 from both sides to get \)3x < 2\(, then divide by 3 to get \)x < \frac{2}{3}$.
5. Combine the two inequalities to write the solution as an interval: \(\frac{-4}{3} < x < \frac{2}{3}\).

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 by coefficients. Proper isolation ensures the inequality can be correctly interpreted and split into two linear inequalities.

Solving Compound Inequalities

When an absolute value inequality is less than a positive number, it translates into a compound inequality combining two inequalities with 'and'. Solving these requires handling both inequalities simultaneously to find the range of values satisfying the original inequality.