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

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 $\varnothing$.)
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Solution

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Solution Steps

To solve the absolute value inequality 2x1+45 |2x - 1| + 4 \leq 5 , we first isolate the absolute value expression by subtracting 4 from both sides, resulting in 2x11 |2x - 1| \leq 1 . This inequality can be split into two separate inequalities: 2x11 2x - 1 \leq 1 and 2x11 2x - 1 \geq -1 . Solving these inequalities will give us the range of values for x x . Finally, we express the solution in interval notation.

Step 1: Understand the Problem

We are given the inequality 2x1+45 |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:

2x1+4454 |2x - 1| + 4 - 4 \leq 5 - 4

This simplifies to:

2x11 |2x - 1| \leq 1

Step 3: Solve the Absolute Value Inequality

The inequality 2x11 |2x - 1| \leq 1 can be split into two separate inequalities:

  1. 2x11 2x - 1 \leq 1
  2. 2x11 2x - 1 \geq -1
Solving the First Inequality

2x11 2x - 1 \leq 1

Add 1 to both sides:

2x2 2x \leq 2

Divide both sides by 2:

x1 x \leq 1

Solving the Second Inequality

2x11 2x - 1 \geq -1

Add 1 to both sides:

2x0 2x \geq 0

Divide both sides by 2:

x0 x \geq 0

Step 4: Combine the Solutions

The solution to the inequality 2x11 |2x - 1| \leq 1 is the intersection of the solutions to the two inequalities:

0x1 0 \leq x \leq 1

Step 5: Write the Solution in Interval Notation

The solution in interval notation is:

[0,1] [0, 1]

Final Answer

The solution to the inequality 2x1+45 |2x - 1| + 4 \leq 5 is [0,1]\boxed{[0, 1]}.

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