Certainly! Let's solve the inequality step-by-step:
1. Given Inequality:
[tex]\[
-4(f-8) \geq 20
\][/tex]
2. Distributing the [tex]\(-4\)[/tex] across the parentheses:
[tex]\[
-4(f - 8) = -4f + 32
\][/tex]
So, the inequality becomes:
[tex]\[
-4f + 32 \geq 20
\][/tex]
3. Isolate the variable [tex]\(f\)[/tex]:
- Subtract 32 from both sides to move the constant term to the right side:
[tex]\[
-4f + 32 - 32 \geq 20 - 32
\][/tex]
Simplify:
[tex]\[
-4f \geq -12
\][/tex]
4. Divide by [tex]\(-4\)[/tex] to solve for [tex]\(f\)[/tex]:
- When dividing by a negative number, remember to reverse the inequality direction:
[tex]\[
\frac{-4f}{-4} \leq \frac{-12}{-4}
\][/tex]
Simplify:
[tex]\[
f \leq 3
\][/tex]
So, the solution to the inequality is:
[tex]\[
f \leq 3
\][/tex]
Next, let's graph the solution on a number line.
1. Draw a number line with appropriate intervals including the point [tex]\( 3 \)[/tex].
2. Mark the endpoint at [tex]\(3\)[/tex]:
- Since [tex]\(f\)[/tex] can be equal to [tex]\(3\)[/tex], we will use a closed circle (●) at [tex]\(3\)[/tex].
3. Shade all values to the left of [tex]\(3\)[/tex]:
- The inequality [tex]\(f \leq 3\)[/tex] includes all values less than or equal to [tex]\(3\)[/tex], so we will shade the line to the left of [tex]\(3\)[/tex] and include the closed circle.
The graphical representation is:
```
<----●=====>
3
```
In this graph, the closed circle at [tex]\(3\)[/tex] indicates that [tex]\(3\)[/tex] is included in the solution and the line extending to the left indicates that all numbers less than [tex]\(3\)[/tex] are also part of the solution.
