Solve each inequality. Give the solution set in interval notation. | 3x - 4 | < 2

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Recognize that the inequality involves an absolute value expression: \(|3x - 4| < 2\). Recall that \(|A| < B\) means \(-B < A < B\) for any real numbers \(A\) and \(B > 0\).

Rewrite the inequality without the absolute value by setting up a compound inequality: \(-2 < 3x - 4 < 2\).

Solve the left part of the compound inequality: \(-2 < 3x - 4\). Add 4 to both sides to isolate the term with \(x\): \(-2 + 4 < 3x\), which simplifies to \(2 < 3x\).

Solve the right part of the compound inequality: \(3x - 4 < 2\). Add 4 to both sides: \(3x < 2 + 4\), which simplifies to \(3x < 6\).

Combine both inequalities and solve for \(x\): from \(2 < 3x < 6\), divide all parts by 3 (since 3 is positive, the inequality signs remain the same), resulting in \(\frac{2}{3} < x < 2\). Express the solution set in interval notation as \((\frac{2}{3}, 2)\).

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Absolute Value Inequalities

Absolute value inequalities involve expressions where the absolute value of a variable or expression is compared to a number. For inequalities like |A| < B, the solution is the set of values for which the expression inside the absolute value lies between -B and B. This concept helps transform the inequality into a compound inequality without absolute values.

Solving Compound Inequalities

Compound inequalities combine two inequalities joined by 'and' or 'or'. For |3x - 4| < 2, it translates to -2 < 3x - 4 < 2. Solving compound inequalities requires isolating the variable by performing algebraic operations on all parts simultaneously to find the range of values satisfying both inequalities.

Interval Notation

Interval notation is a concise way to represent solution sets of inequalities using parentheses and brackets. Parentheses indicate that endpoints are not included, while brackets mean they are included. For example, (a, b) represents all numbers between a and b, excluding a and b, which is essential for expressing the solution set clearly.