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Chapter 0: Problem 83

Solve each absolute value inequality. $$3 \leq|2 x-1|$$

Short Answer

Expert verified
The solution to the absolute value inequality is \(x \geq 2\).

Step by step solution

01

Identify the absolute value equation

In this exercise, the absolute value inequality is \(3 \leq|2 x-1|\).
02

Split into two inequalities

An absolute-value inequality can be split into two inequalities - one representing the positive solution and the other the negative. This can be represented as: \(3 \leq 2x - 1 \) and \(3 \leq -(2x - 1)\).
03

Solve the first inequality

Start by solving the first inequality: \(3 \leq 2x - 1\) Add 1 to both sides to isolate the term with x: \(4 \leq 2x\) Then divide each side by 2, to solve for x: \(x \geq 2\).
04

Solve the second inequality

Now, solve the second inequality: \(3 \leq -(2x - 1)\). Start by distributing the negative sign: \(3 \leq -2x +1\). Subtract 1 from both sides: \(2 \leq -2x\). Then divide each side by -2. Since we are dividing by a negative number, the direction of the inequality changes: \(x \geq -1\).
05

Obtain the final solution

Combining the solutions from the two inequalities, we get \(x \geq 2\) and \(x \geq -1\). However, \(x \geq 2\) encompasses \(x \geq -1\) so the solution to the inequality is \(x \geq 2\).

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