Sure, let's graph the compound inequality [tex]\( x > -1 \)[/tex] and [tex]\( x < 4 \)[/tex] on the number line. Follow these steps:
1. Draw the number line: Start by drawing a horizontal line and marking the points corresponding to [tex]\(-1\)[/tex] and [tex]\(4\)[/tex].
2. Plot [tex]\( x > -1 \)[/tex]:
- Identify the point [tex]\(-1\)[/tex] on the number line.
- Since the inequality is strictly greater than [tex]\(-1\)[/tex] (i.e., [tex]\( x > -1 \)[/tex] and not [tex]\( x \ge -1 \)[/tex]), we use an open circle at [tex]\(-1\)[/tex] to indicate that [tex]\(-1\)[/tex] is not included.
- Shade the region to the right of [tex]\(-1\)[/tex] to show that all numbers greater than [tex]\(-1\)[/tex] satisfy this part of the inequality.
3. Plot [tex]\( x < 4 \)[/tex]:
- Identify the point [tex]\(4\)[/tex] on the number line.
- Since the inequality is strictly less than [tex]\(4\)[/tex] (i.e., [tex]\( x < 4 \)[/tex] and not [tex]\( x \le 4 \)[/tex]), we use an open circle at [tex]\(4\)[/tex] to indicate that [tex]\(4\)[/tex] is not included.
- Shade the region to the left of [tex]\(4\)[/tex] to show that all numbers less than [tex]\(4\)[/tex] satisfy this part of the inequality.
4. Combine the inequalities:
- The compound inequality [tex]\( x > -1 \text{ and } x < 4 \)[/tex] implies that [tex]\( x \)[/tex] must satisfy both conditions simultaneously.
- Therefore, the solution set is the intersection of the two shaded regions, which is the region between [tex]\(-1\)[/tex] and [tex]\(4\)[/tex] but not including [tex]\(-1\)[/tex] and [tex]\(4\)[/tex].
5. Final graph on the number line:
- On the number line, draw an open circle at [tex]\(-1\)[/tex] and another open circle at [tex]\(4\)[/tex].
- Shade the region between these two open circles.
The resulting number line should look like this:
```
<----o=================o---->
-1 4
```
The open circles indicate that the endpoints [tex]\(-1\)[/tex] and [tex]\(4\)[/tex] are not included in the solution set, and the shaded region between the circles represents all numbers that satisfy the compound inequality [tex]\( x > -1 \text{ and } x < 4 \)[/tex].
