Absolute Value Inequalities Math Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

medium
Solve ∣2x−1∣≥3|2x - 1| \geq 3.

Solution

  1. 1
    ∣A∣≥k|A| \geq k means A≥kA \geq k OR A≤−kA \leq -k.
  2. 2
    Case 1: 2x−1≥3⇒2x≥4⇒x≥22x - 1 \geq 3 \Rightarrow 2x \geq 4 \Rightarrow x \geq 2.
  3. 3
    Case 2: 2x−1≤−3⇒2x≤−2⇒x≤−12x - 1 \leq -3 \Rightarrow 2x \leq -2 \Rightarrow x \leq -1.
  4. 4
    Solution: x≤−1x \leq -1 or x≥2x \geq 2. In interval notation: (−∞,−1]∪[2,∞)(-\infty, -1] \cup [2, \infty).

Answer

x≤−1 or x≥2x \leq -1 \text{ or } x \geq 2
For ∣A∣≥k|A| \geq k (greater than or equal), the solution is two separate intervals. The expression is 'far from zero,' meaning it is on the outer parts of the number line.

About Absolute Value Inequalities

Absolute value inequalities describe values within or outside a fixed distance from a center.

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More Absolute Value Inequalities Examples

Example 1 easy

Solve [formula].

Example 3 easy

Solve [formula].

Example 4 hard

Solve [formula].

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