In Exercises 59–94, solve each absolute value inequality. 1 < |2 - 3x|

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Recall that the inequality \(1 < |2 - 3x|\) means the distance between \(2 - 3x\) and 0 is greater than 1.

Rewrite the inequality \(1 < |2 - 3x|\) as two separate inequalities: \(2 - 3x < -1\) or \(2 - 3x > 1\).

Solve the first inequality \(2 - 3x < -1\) by isolating \(x\): subtract 2 from both sides to get \(-3x < -3\), then divide both sides by \(-3\) (remember to reverse the inequality sign when dividing by a negative number).

Solve the second inequality \(2 - 3x > 1\) by isolating \(x\): subtract 2 from both sides to get \(-3x > -1\), then divide both sides by \(-3\) (again, reverse the inequality sign).

Combine the solutions from both inequalities to express the solution set for \(x\) that satisfies \(1 < |2 - 3x|\).

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주요 개념

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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

To solve inequalities involving absolute values, such as |A| > c, we split them into two separate inequalities: A > c or A < -c when c is positive. This approach transforms the absolute value inequality into a compound inequality that can be solved using standard algebraic methods.

Compound Inequalities

Compound inequalities involve two inequalities combined by 'and' or 'or'. For absolute value inequalities, solutions often form two intervals combined by 'or'. Understanding how to solve and graph compound inequalities is essential for interpreting the solution set correctly.