Question
Solve the inequality. Write the solution set in interval notation. \( 11 \leq 6(n+4)-4 n \)
Ask by Sanders Bush. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The solution is \( n \geq -\frac{13}{2} \), which in interval notation is \(\left[ -\frac{13}{2}, \infty \right)\).
Solution
To solve the inequality \( 11 \leq 6(n+4) - 4n \):
1. **Expand and simplify the right-hand side:**
\[
6(n + 4) - 4n = 6n + 24 - 4n = 2n + 24
\]
2. **Rewrite the inequality:**
\[
11 \leq 2n + 24
\]
3. **Isolate \( n \):**
\[
11 - 24 \leq 2n \quad \Rightarrow \quad -13 \leq 2n
\]
4. **Divide both sides by 2:**
\[
-\frac{13}{2} \leq n
\]
5. **Express the solution in interval notation:**
\[
\left[ -\frac{13}{2}, \infty \right)
\]
**Final Answer:**
\(\left[ -\dfrac{13}{2},\ \infty\, \right)\)
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Beyond the Answer
Let's first simplify the inequality. Starting with: \[ 11 \leq 6(n+4) - 4n \] Distributing the 6 gives: \[ 11 \leq 6n + 24 - 4n \] Now, combine like terms: \[ 11 \leq 2n + 24 \] Next, we'll isolate \( n \) by subtracting 24 from both sides: \[ 11 - 24 \leq 2n \] \[ -13 \leq 2n \] Now, divide both sides by 2: \[ -\frac{13}{2} \leq n \] This can also be written as: \[ n \geq -\frac{13}{2} \] In interval notation, the solution set is: \[ \left[-\frac{13}{2}, \infty\right) \]
