Let's solve the compound inequality step by step:
The compound inequality to graph is:
[tex]\[ x \geq -5 \text{ and } x \leq 4 \][/tex]
### Step 1: Understand the Inequality Components
- [tex]\( x \geq -5 \)[/tex] means [tex]\( x \)[/tex] is greater than or equal to [tex]\(-5\)[/tex].
- [tex]\( x \leq 4 \)[/tex] means [tex]\( x \)[/tex] is less than or equal to [tex]\(4\)[/tex].
### Step 2: Identify the Range of Solutions
The solutions to the compound inequality are all the values of [tex]\( x \)[/tex] that satisfy both parts simultaneously. Therefore, we combine these two conditions to state:
[tex]\[ -5 \leq x \leq 4 \][/tex]
### Step 3: Graph on a Number Line
1. Draw a horizontal line to represent the number line.
2. Place a closed circle at [tex]\(-5\)[/tex] to indicate that [tex]\(-5\)[/tex] is included in the solution set (because of the "greater than or equal to" condition).
3. Place a closed circle at [tex]\(4\)[/tex] to indicate that [tex]\(4\)[/tex] is included in the solution set (because of the "less than or equal to" condition).
4. Shade the region on the number line between [tex]\(-5\)[/tex] and [tex]\(4\)[/tex], inclusive of both endpoints.
Here's how the graph should look:
[tex]\[ \below 3 { \text{Number Line:} \ \ -5 \, \bigcirc \haprmline{-5, 4} \, \bigcirc \, 4 \,} \][/tex]
This represents all the numbers [tex]\( x \)[/tex] such that:
[tex]\[ -5 \leq x \leq 4 \][/tex]
Therefore, the shaded region between [tex]\(-5\)[/tex] and [tex]\(4\)[/tex] including the end points, is the graphical representation of the compound inequality [tex]\( x \geq -5 \text{ and } x \leq 4 \)[/tex].
