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

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Identify the inequality involving the absolute value: \(|3x - 4| \geq 2\).

Recall that for an inequality of the form \(|A| \geq B\) (where \(B > 0\)), the solution splits into two cases: \(A \geq B\) or \(A \leq -B\).

Set up the two inequalities based on the definition: \(3x - 4 \geq 2\) and \(3x - 4 \leq -2\).

Solve each inequality separately: - For \(3x - 4 \geq 2\), add 4 to both sides to get \(3x \geq 6\), then divide both sides by 3 to find \(x \geq 2\). - For \(3x - 4 \leq -2\), add 4 to both sides to get \(3x \leq 2\), then divide both sides by 3 to find \(x \leq \frac{2}{3}\).

Combine the two solution sets to express the final answer in interval notation: \((-\infty, \frac{2}{3}] \cup [2, \infty)\).

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Key Concepts

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

Absolute Value Inequalities

Absolute value inequalities involve expressions within absolute value bars, representing distance from zero. To solve |A| ≥ B, where B is positive, split into two cases: A ≥ B or A ≤ -B. This approach helps convert the inequality into simpler linear inequalities.

Solving Linear Inequalities

Linear inequalities are solved by isolating the variable on one side using algebraic operations like addition, subtraction, multiplication, or division. When multiplying or dividing by a negative number, the inequality sign reverses. Solutions are expressed as ranges or intervals.

Interval Notation

Interval notation is a concise way to represent solution sets of inequalities using parentheses and brackets. Parentheses indicate values not included (open interval), while brackets indicate inclusion (closed interval). It clearly shows the range of values satisfying the inequality.

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