Question

Solve the inequality both algebraically and graphically. Draw a number line graph of the solution and give interval notation.
11xminus9less than or equals9plus5x

Answer 1

To solve the inequality \( 11x - 9 \leq 9 + 5x \) algebraically, we will follow these steps:

1. **Rearrange the inequality**:
Subtract \( 5x \) from both sides:
\[
11x - 5x - 9 \leq 9
\]
This simplifies to:
\[
6x - 9 \leq 9
\]

2. **Isolate the variable**:
Add \( 9 \) to both sides:
\[
6x \leq 18
\]

Now, divide both sides by \( 6 \):
\[
x \leq 3
\]

Thus, the solution to the inequality is \( x \leq 3 \).

### Graphical Solution
To represent this solution graphically on a number line:

- Draw a horizontal number line.
- Mark the point \( 3 \) on the number line.
- Since the inequality is inclusive (the \( \leq \) symbol), shade the region to the left of \( 3 \) and include the point \( 3 \) with a closed dot.

### Interval Notation
The interval notation for the solution \( x \leq 3 \) is:
\[
(-\infty, 3]
\]

### Summary
- **Algebraic Solution**: \( x \leq 3 \)
- **Graphical Solution**: The number line is shaded to the left of \( 3 \), including \( 3 \).
- **Interval Notation**: \( (-\infty, 3] \)