Question

Solve the inequality both algebraically and graphically. Draw a number line graph of the solution and give interval notation.

Answer 1

To solve an inequality, I can provide a general example. Let's solve a common inequality: \(2x - 5 < 3\).

### Step 1: Solve Algebraically

1. **Add 5 to both sides:**
\[
2x - 5 + 5 < 3 + 5
\]
\[
2x < 8
\]

2. **Divide by 2:**
\[
x < 4
\]

### Step 2: Interval Notation
The solution to the inequality \(x < 4\) in interval notation is:
\[
(-\infty, 4)
\]

### Step 3: Graphically
To represent this on a number line:

1. Draw a horizontal line and mark a point for 4 on it.
2. Since the inequality is strict (\(<\)), use an open circle at 4 to indicate that 4 is not included in the solution.
3. Shade the line to the left of 4 to indicate that all numbers less than 4 are included in the solution.

Here's how the number line would look:

```
<---|---|---|---|---|---|---|---|---|---|---|---|---|--->
-5 -4 -3 -2 -1 0 1 2 3 4 5 6
(o)
```
The point labeled (o) indicates an open circle at 4, with shading extending leftwards toward negative infinity.

### Conclusion
The solution to the inequality \(2x - 5 < 3\) is:
- **Algebraically:** \(x < 4\)
- **Interval notation:** \((- \infty, 4)\)
- **Graphically:** A number line showing shading to the left of the open circle at 4.