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≤5, we first isolate the absolute value expression by subtracting 4 from both sides, resulting in ∣2x−1∣≤1. This inequality can be split into two separate inequalities: 2x−1≤1 and 2x−1≥−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≤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≤5−4
This simplifies to:
∣2x−1∣≤1
Step 3: Solve the Absolute Value Inequality
The inequality ∣2x−1∣≤1 can be split into two separate inequalities:
2x−1≤1
2x−1≥−1
Solving the First Inequality
2x−1≤1
Add 1 to both sides:
2x≤2
Divide both sides by 2:
x≤1
Solving the Second Inequality
2x−1≥−1
Add 1 to both sides:
2x≥0
Divide both sides by 2:
x≥0
Step 4: Combine the Solutions
The solution to the inequality ∣2x−1∣≤1 is the intersection of the solutions to the two inequalities:
0≤x≤1
Step 5: Write the Solution in Interval Notation
The solution in interval notation is:
[0,1]
Final Answer
The solution to the inequality ∣2x−1∣+4≤5 is [0,1].