To solve the inequality [tex]\(\frac{1}{4}(x - 3) \leq -2\)[/tex] and express the solution graphically, follow these steps:
1. Isolate [tex]\(x\)[/tex] in the inequality:
Start with:
[tex]\[
\frac{1}{4}(x - 3) \leq -2
\][/tex]
To eliminate the fraction, multiply both sides of the inequality by 4:
[tex]\[
4 \cdot \frac{1}{4}(x - 3) \leq 4 \cdot (-2)
\][/tex]
Simplify to get:
[tex]\[
x - 3 \leq -8
\][/tex]
2. Solve for [tex]\(x\)[/tex]:
Add 3 to both sides to isolate [tex]\(x\)[/tex]:
[tex]\[
x - 3 + 3 \leq -8 + 3
\][/tex]
Simplify the inequality:
[tex]\[
x \leq -5
\][/tex]
Thus, the solution to the inequality is:
[tex]\[
x \leq -5
\][/tex]
3. Express the solution graphically:
To graph this solution on a number line:
- Draw a number line with appropriate scale.
- Locate the point [tex]\(-5\)[/tex] on the number line.
- Since the inequality is [tex]\(\leq\)[/tex], place a solid dot at [tex]\(-5\)[/tex] to indicate that [tex]\(-5\)[/tex] is included in the solution.
- Shade the region to the left of [tex]\(-5\)[/tex], extending to negative infinity, to represent all values less than or equal to [tex]\(-5\)[/tex].
Graphically, it looks like this:
```
<---|---|---|---|---|---|---|---|---|---|---|--->
-8 -7 -6 -5 -4 -3 -2
•===============>
```
The solid dot at [tex]\(-5\)[/tex] and the shading to the left indicate that any value of [tex]\(x\)[/tex] less than or equal to [tex]\(-5\)[/tex] satisfies the inequality [tex]\(\frac{1}{4}(x - 3) \leq -2\)[/tex].
