To solve the inequality [tex]\(\frac{1}{4}(x-3) \leq -2\)[/tex] and express the solution graphically, let's follow these detailed steps:
### Step 1: Solve the Inequality Algebraically
1. Original Inequality:
[tex]\[\frac{1}{4}(x-3) \leq -2\][/tex]
2. Clear the fraction:
Multiply both sides by 4 to eliminate the fraction:
[tex]\[x - 3 \leq -2 \times 4\][/tex]
Simplify:
[tex]\[x - 3 \leq -8\][/tex]
3. Solve for [tex]\(x\)[/tex]:
Add 3 to both sides:
[tex]\[x \leq -8 + 3\][/tex]
Simplify:
[tex]\[x \leq -5\][/tex]
So, the algebraic solution is:
[tex]\[x \leq -5\][/tex]
### Step 2: Graphical Representation
To display this solution graphically:
1. Number Line:
Draw a number line showing the point [tex]\(x = -5\)[/tex].
2. Indicate the Solution:
- The solution [tex]\(x \leq -5\)[/tex] means all values of [tex]\(x\)[/tex] that are less than or equal to [tex]\(-5\)[/tex].
- Represent this by shading or coloring the region to the left of [tex]\(-5\)[/tex] on the number line.
- Use a solid circle or dot at [tex]\(x = -5\)[/tex] to indicate that [tex]\(-5\)[/tex] is included in the solution (because of the ≤ symbol).
### Step 3: Example Graph
Below is how the number line would look:
[tex]\[ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad | \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \][/tex]
[tex]\[ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad -5 \][/tex]
Sketch:
```
<===●------------->
-5
```
- Here, the solid circle (●) at [tex]\(-5\)[/tex] indicates that [tex]\(-5\)[/tex] is part of the solution.
- The arrow pointing to the left signifies that all values less than [tex]\(-5\)[/tex] are also part of the solution.
### Visualization
In summary, the graph of the solution to the inequality [tex]\(\frac{1}{4}(x-3) \leq -2\)[/tex] is a number line with:
1. A solid circle (or dot) at [tex]\(x = -5\)[/tex].
2. A shaded region extending to the left from [tex]\(x = -5\)[/tex], representing all [tex]\(x\)[/tex] values less than or equal to [tex]\(-5\)[/tex].
