Question
Solve the inequality involving absolute value. Write your final answer in interval notation. (If the solution set is empty, enter EMPTY or \( \varnothing \).) \[ |2 x+1|+1 \leq 8 \] +
Ask by Whittaker Ryan. in the United States
Mar 10,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The solution is all numbers \( x \) between \(-4\) and \(3\), inclusive. In interval notation, this is \([-4, 3]\).
Solution
We begin with the inequality
\[
|2x+1|+1 \leq 8.
\]
**Step 1: Isolate the absolute value expression.**
Subtract \(1\) from both sides:
\[
|2x+1| \leq 7.
\]
**Step 2: Rewrite the absolute value inequality as a compound inequality.**
Recall that for any expression \(A\), if \(|A| \leq B\) (with \(B \geq 0\)), then
\[
-B \leq A \leq B.
\]
Here, \(A = 2x+1\) and \(B = 7\). Thus, we have:
\[
-7 \leq 2x+1 \leq 7.
\]
**Step 3: Solve the compound inequality.**
Subtract \(1\) from all parts:
\[
-7 - 1 \leq 2x+1 - 1 \leq 7 - 1,
\]
which simplifies to:
\[
-8 \leq 2x \leq 6.
\]
Divide the entire inequality by \(2\) (since \(2\) is positive, the inequality directions remain the same):
\[
-4 \leq x \leq 3.
\]
**Final Answer:**
The solution set in interval notation is:
\[
[-4, 3].
\]
Answered by UpStudy AI and reviewed by a Professional Tutor
Like
error msg
Explain 
Simplify this solution The Deep Dive
To solve the inequality \( |2x + 1| + 1 \leq 8 \), we first isolate the absolute value: \[ |2x + 1| \leq 8 - 1 \] \[ |2x + 1| \leq 7 \] This means we can split it into two separate inequalities: 1. \( 2x + 1 \leq 7 \) 2. \( 2x + 1 \geq -7 \) Now, let's solve each inequality: 1. For \( 2x + 1 \leq 7 \): \[ 2x \leq 6 \quad \Rightarrow \quad x \leq 3 \] 2. For \( 2x + 1 \geq -7 \): \[ 2x \geq -8 \quad \Rightarrow \quad x \geq -4 \] Combining these results, we find: \(-4 \leq x \leq 3\) In interval notation, the solution is: \([-4, 3]\)
