Ch. 1 - Equations and Inequalities

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo

Blitzer 8th Edition

Ch. 1 - Equations and Inequalities

Problema 80a

Capitolo 2, Problema 80a

Solve each absolute value inequality. 5|2x + 1| - 3 ≥ 9

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Rewrite the inequality to isolate the absolute value expression. Start by adding 3 to both sides: 5|2x + 1| ≥ 12.

Divide both sides of the inequality by 5 to further isolate the absolute value: |2x + 1| ≥ 12/5.

Recall the definition of absolute value inequalities. For |A| ≥ B, the inequality splits into two cases: A ≥ B or A ≤ -B. Apply this to the expression: 2x + 1 ≥ 12/5 or 2x + 1 ≤ -12/5.

Solve each inequality separately. For the first case, subtract 1 from both sides: 2x ≥ 12/5 - 5/5. Then divide by 2: x ≥ (12/5 - 5/5)/2. For the second case, subtract 1 from both sides: 2x ≤ -12/5 - 5/5. Then divide by 2: x ≤ (-12/5 - 5/5)/2.

Combine the solutions from both cases to express the final solution set. The solution will be in the form of two intervals: x ≥ (12/5 - 5/5)/2 or x ≤ (-12/5 - 5/5)/2.

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Absolute Value

Absolute value represents the distance of a number from zero on the number line, regardless of direction. For any real number x, the absolute value is denoted as |x|, which equals x if x is non-negative and -x if x is negative. Understanding absolute value is crucial for solving inequalities that involve expressions within absolute value symbols.

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Inequalities

Inequalities express a relationship between two expressions that are not necessarily equal. They can be represented using symbols such as >, <, ≥, or ≤. Solving inequalities often involves finding the range of values that satisfy the condition, which can include multiple solutions or intervals, especially when absolute values are involved.

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Solving Absolute Value Inequalities

To solve an absolute value inequality, one must consider the definition of absolute value and break the inequality into two separate cases. For example, the inequality |A| ≥ B leads to two scenarios: A ≥ B or A ≤ -B. This approach allows for finding all possible solutions that satisfy the original inequality, which is essential for complete problem-solving.

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