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.
7(n+2​)−3n≤4​(n+7​)

Answer 1

To solve the inequality \( 7(n+2) - 3n \leq 4(n+7) \), we will follow these steps:

1. **Distribute** on both sides.
2. **Combine like terms**.
3. **Isolate \( n \)**.
4. **Express the solution in set-builder notation and interval notation**.

### Step 1: Distributing
Distributing on both sides gives us:
\[
7n + 14 - 3n \leq 4n + 28.
\]

### Step 2: Combine like terms
Combining the \( n \) terms on the left side:
\[
(7n - 3n) + 14 \leq 4n + 28,
\]
which simplifies to:
\[
4n + 14 \leq 4n + 28.
\]

### Step 3: Isolating \( n \)
Subtract \( 4n \) from both sides:
\[
14 \leq 28.
\]
This statement is always true (since \( 14 \) is indeed less than \( 28 \)).

### Conclusion
Since the inequality \( 14 \leq 28 \) is always true, this implies that any real number \( n \) satisfies the inequality.

### Set-Builder Notation
The solution set in set-builder notation is:
\[
\{ n \in \mathbb{R} \}.
\]

### Interval Notation
The solution set in interval notation is:
\[
(-\infty, \infty).
\]

### Graphing the Solution Set
On the real number line, this solution means that:

- All points on the line are included.
- You can simply draw a line that continues infinitely in both directions.

This completes our solution to the inequality.