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

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.

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

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:

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

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

1. \( 2x - 1 \leq 1 \)
2. \( 2x - 1 \geq -1 \)

\[ 2x - 1 \leq 1 \]

Add 1 to both sides:

\[ 2x \leq 2 \]

Divide both sides by 2:

\[ x \leq 1 \]

\[ 2x - 1 \geq -1 \]

Add 1 to both sides:

\[ 2x \geq 0 \]

Divide both sides by 2:

\[ x \geq 0 \]

The solution to the inequality \( |2x - 1| \leq 1 \) is the intersection of the solutions to the two inequalities:

\[ 0 \leq x \leq 1 \]

The solution in interval notation is:

\[ [0, 1] \]

Final Answer