Questions: Solve 2x-1+4 ≤ 5 and write the solution in interval notation. (If there is no solution, enter your answer as ∅.)

Transcript text: Solve $|2 x-1|+4 \leq 5$ and write the solution in interval notation. (If there is no solution, enter your answer as $\varnothing$.)

Solution

Solution Steps

To solve the absolute value inequality $ |2x - 1| + 4 \leq 5 $, we first isolate the absolute value expression by subtracting 4 from both sides, resulting in $ |2x - 1| \leq 1 $. This inequality can be split into two separate inequalities: $ 2x - 1 \leq 1 $ and $ 2x - 1 \geq -1 $. Solving these inequalities will give us the range of values for $ x $. Finally, we express the solution in interval notation.

Step 1: Understand the Problem

We are given the inequality $ |2x - 1| + 4 \leq 5 $. Our task is to solve this inequality and express the solution in interval notation.

Step 2: Isolate the Absolute Value

First, we need to isolate the absolute value expression. We do this by subtracting 4 from both sides of the inequality:

\[ |2x - 1| + 4 - 4 \leq 5 - 4 \]

This simplifies to:

\[ |2x - 1| \leq 1 \]

Step 3: Solve the Absolute Value Inequality

The inequality $ |2x - 1| \leq 1 $ can be split into two separate inequalities: