Question
Solve \( |x-7| \leq 4 \) State your answer as a compound inequality \( A \leq x \leq B \) : State your answer in interval notation \( [A, B] \) :
Ask by Davey Murphy. in the United States
Feb 02,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( 3 \leq x \leq 11 \) or \([3, 11]\).
Solution
To solve the inequality \( |x - 7| \leq 4 \), we can interpret it as the distance between \( x \) and 7 being at most 4. This can be rewritten as a compound inequality and also expressed in interval notation.
### Step-by-Step Solution:
1. **Start with the absolute inequality:**
\[
|x - 7| \leq 4
\]
2. **Rewrite the absolute inequality as a compound inequality:**
\[
-4 \leq x - 7 \leq 4
\]
3. **Isolate \( x \) by adding 7 to all parts of the inequality:**
\[
-4 + 7 \leq x \leq 4 + 7
\]
\[
3 \leq x \leq 11
\]
### Final Answers:
- **Compound Inequality:**
\[
3 \leq x \leq 11
\]
- **Interval Notation:**
\[
[3,\, 11]
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Bonus Knowledge
To solve \( |x-7| \leq 4 \), we can break it into two parts based on the definition of absolute value. This means we have: 1. \( x - 7 \leq 4 \) 2. \( x - 7 \geq -4 \) Starting with the first inequality: 1. \( x - 7 \leq 4 \) - Adding 7 to both sides gives us \( x \leq 11 \). Now for the second inequality: 2. \( x - 7 \geq -4 \) - Adding 7 to both sides gives us \( x \geq 3 \). Putting this together, we have the compound inequality: \[ 3 \leq x \leq 11 \] In interval notation, it is expressed as: \[ [3, 11] \]
