Question
Solve: \( |2 x+7|<8 \) Give your answer using interval notation.
Ask by Hart Young. in the United States
Feb 02,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The solution is \( \left( -\frac{15}{2}, \frac{1}{2} \right) \).
Solution
To solve the inequality \( |2x + 7| < 8 \), follow these steps:
1. **Understand the Absolute Value Inequality:**
\[
|2x + 7| < 8 \quad \text{means} \quad -8 < 2x + 7 < 8
\]
2. **Solve the Compound Inequality:**
- Subtract 7 from all parts:
\[
-8 - 7 < 2x < 8 - 7 \quad \Rightarrow \quad -15 < 2x < 1
\]
- Divide by 2:
\[
\frac{-15}{2} < x < \frac{1}{2}
\]
3. **Write the Solution in Interval Notation:**
\[
\left( -\frac{15}{2}, \frac{1}{2} \right)
\]
**Answer:** \(\left( -\dfrac{15}{2},\ \dfrac{1}{2} \right)\)
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Simplify this solution The Deep Dive
To solve the inequality \( |2x + 7| < 8 \), we start by breaking it into two separate cases based on the definition of absolute value. 1. \( 2x + 7 < 8 \) 2. \( 2x + 7 > -8 \) For the first inequality \( 2x + 7 < 8 \): Subtract 7 from both sides: \[ 2x < 1 \] Divide both sides by 2: \[ x < \frac{1}{2} \] For the second inequality \( 2x + 7 > -8 \): Subtract 7 from both sides: \[ 2x > -15 \] Divide both sides by 2: \[ x > -\frac{15}{2} \] Combining the two inequalities, we get: \[ -\frac{15}{2} < x < \frac{1}{2} \] In interval notation, the solution is: \[ \left(-\frac{15}{2}, \frac{1}{2}\right) \]
