Question
Solve the linear inequality. Express the solution using interval notation. \[ 1<3 x+4 \leq 28 \]
Ask by French Simmons. in the United States
Jan 28,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The solution is \( -1 < x \leq 8 \), which in interval notation is \( (-1, 8] \).
Solution
To solve the linear inequality \( 1 < 3x + 4 \leq 28 \), follow these steps:
1. **Subtract 4 from all parts of the inequality:**
\[
1 - 4 < 3x + 4 - 4 \leq 28 - 4
\]
\[
-3 < 3x \leq 24
\]
2. **Divide all parts by 3 to solve for \( x \):**
\[
\frac{-3}{3} < x \leq \frac{24}{3}
\]
\[
-1 < x \leq 8
\]
3. **Express the solution in interval notation:**
\[
(-1, 8]
\]
**Final Answer:**
\[
(-1,8]
\]
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Simplify this solution Bonus Knowledge
To solve the compound inequality \( 1 < 3x + 4 \leq 28 \), we can break it down into two parts. First, take the lower part of the inequality: 1. Subtract 4 from both sides: \[ 1 - 4 < 3x \] \[ -3 < 3x \] Then, divide by 3: \[ -1 < x \] or \( x > -1 \). Next, tackle the upper part of the inequality: 2. From \( 3x + 4 \leq 28 \), subtract 4 from both sides: \[ 3x \leq 28 - 4 \] \[ 3x \leq 24 \] Divide by 3: \[ x \leq 8 \]. Now, combining both parts, we have: \[ -1 < x \leq 8 \]. Expressing this in interval notation, the solution is: \[ (-1, 8] \].
