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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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.

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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