Question
\( { }_{\text {Find the solution set of }|2 x-3|+1=6} \)
Ask by Love Wood. in the United States
Jan 29,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The solution set is \(\{-1, 4\}\).
Solution
To solve the equation \( |2x - 3| + 1 = 6 \), follow these steps:
1. **Isolate the Absolute Value:**
\[
|2x - 3| + 1 = 6 \implies |2x - 3| = 5
\]
2. **Solve the Absolute Value Equation:**
The equation \( |2x - 3| = 5 \) splits into two cases:
- **Case 1:** \( 2x - 3 = 5 \)
\[
2x - 3 = 5 \implies 2x = 8 \implies x = 4
\]
- **Case 2:** \( 2x - 3 = -5 \)
\[
2x - 3 = -5 \implies 2x = -2 \implies x = -1
\]
3. **Solution Set:**
Combining both cases, the solution set is:
\[
\{\, -1,\, 4 \,\}
\]
**Final Answer:**
\[
\boxed{\,\left\{\, -1,\;\,4\,\right\}\,}
\]
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Simplify this solution The Deep Dive
To solve the equation \( |2x - 3| + 1 = 6 \), we first subtract 1 from both sides, giving us \( |2x - 3| = 5 \). This absolute value equation breaks down into two cases: Case 1: \( 2x - 3 = 5 \) leads to \( 2x = 8 \) or \( x = 4 \). Case 2: \( 2x - 3 = -5 \) leads to \( 2x = -2 \) or \( x = -1 \). So, the solution set is \( \{ -1, 4 \} \). If you've ever faced confusion with absolute values, remember that they represent the distance from zero, which can lead to two possible equations. Always break it down for clarity!
