Question
Find the solution set for the inequality. \( |q+5| \leq 7 \) Select the correct choice below and fill in any answer boxes present in your choice. A. The solution set is \( \{q \square\} \). (Type an inequality or a compound inequality.) B. The solution is the set of all real numbers. C. The solution is the empty set.
Ask by Nichols Bates. in the United States
Feb 04,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The solution set is all real numbers \( q \) where \( -12 \leq q \leq 2 \).
Solution
To solve the inequality \( |q + 5| \leq 7 \), follow these steps:
1. **Understand the Absolute Value Inequality:**
The inequality \( |x| \leq a \) means that the expression inside the absolute value is between \(-a\) and \(a\).
2. **Apply This to the Given Inequality:**
\[
|q + 5| \leq 7 \implies -7 \leq q + 5 \leq 7
\]
3. **Solve for \( q \):**
Subtract 5 from all parts of the inequality:
\[
-7 - 5 \leq q \leq 7 - 5 \implies -12 \leq q \leq 2
\]
4. **Express the Solution Set:**
The solution set consists of all real numbers \( q \) such that \( -12 \leq q \leq 2 \).
**Correct Choice:**
**A.** The solution set is \( \{ q \mid -12 \leq q \leq 2 \} \).
Answer:
A. The solution set is \( \{q \mid -12 \leq q \leq 2\} \).
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Simplify this solution Mind Expander
To solve the inequality \( |q+5| \leq 7 \), we start by interpreting the absolute value. This means that the expression inside can be between \(-7\) and \(7\). Therefore, we can rewrite the inequality as: \[ -7 \leq q + 5 \leq 7 \] Next, we subtract \(5\) from all parts of the inequality: \[ -7 - 5 \leq q \leq 7 - 5 \] \[ -12 \leq q \leq 2 \] Thus, the solution set is: A. The solution set is \( \{q \mid -12 \leq q \leq 2\} \).
