Question

Solve the inequality and express the solution set in​ set-builder notation and interval notation. Graph the solution set on the real number line.
6(n+2)-2n≤4(n+7​)

Answer 1

To solve the inequality \( 6(n + 2) - 2n \leq 4(n + 7) \), we will follow these steps:

1. **Distribute and simplify both sides:**
\[
6(n + 2) = 6n + 12
\]
So, the left side becomes:
\[
6n + 12 - 2n = 4n + 12
\]

The right side:
\[
4(n + 7) = 4n + 28
\]

Therefore, the inequality now looks like:
\[
4n + 12 \leq 4n + 28
\]

2. **Subtract \( 4n \) from both sides:**
\[
12 \leq 28
\]

This statement is true and does not depend on \( n \).

3. **Conclusion:**
Since \( 12 \leq 28 \) is always true, the solution set includes all real numbers.

### Solution Set in Set-Builder Notation:
The solution can be expressed in set-builder notation as:
\[
\{ n \mid n \in \mathbb{R} \}
\]

### Solution Set in Interval Notation:
The solution can be expressed in interval notation as:
\[
(-\infty, \infty)
\]

### Graphing the Solution Set:
To graph this solution on the real number line, you would draw a line that extends infinitely in both directions, indicating that every real number is included in the solution.

### Summary:
- The solution in set-builder notation is \( \{ n \mid n \in \mathbb{R} \} \).
- The solution in interval notation is \( (-\infty, \infty) \).
- The graph represents all real numbers on the line.