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