### Part A: Solve the Inequality
We start with the given inequality:
[tex]\[
\left| \frac{1}{4}x - 2 \right| - 3 \geq 4
\][/tex]
Step 1: Isolate the absolute value expression.
[tex]\[
\left| \frac{1}{4} x - 2 \right| - 3 \geq 4
\][/tex]
Add 3 to both sides:
[tex]\[
\left| \frac{1}{4} x - 2 \right| \geq 7
\][/tex]
Step 2: Translate the absolute value inequality to two separate inequalities.
The absolute value inequality [tex]\(\left| A \right| \geq B\)[/tex] translates to:
[tex]\[
A \geq B \quad \text{or} \quad A \leq -B
\][/tex]
In our case, [tex]\(A = \frac{1}{4} x - 2\)[/tex] and [tex]\(B = 7\)[/tex]. Therefore, we write:
[tex]\[
\frac{1}{4} x - 2 \geq 7 \quad \text{or} \quad \frac{1}{4} x - 2 \leq -7
\][/tex]
Step 3: Solve each inequality separately.
First inequality:
[tex]\[
\frac{1}{4} x - 2 \geq 7
\][/tex]
Add 2 to both sides:
[tex]\[
\frac{1}{4} x \geq 9
\][/tex]
Multiply both sides by 4:
[tex]\[
x \geq 36
\][/tex]
Second inequality:
[tex]\[
\frac{1}{4} x - 2 \leq -7
\][/tex]
Add 2 to both sides:
[tex]\[
\frac{1}{4} x \leq -5
\][/tex]
Multiply both sides by 4:
[tex]\[
x \leq -20
\][/tex]
Step 4: Combine the solutions from the two inequalities.
[tex]\[
x \geq 36 \quad \text{or} \quad x \leq -20
\][/tex]
So, the solution to the inequality is:
[tex]\[
x \geq 36 \quad \text{or} \quad x \leq -20
\][/tex]
### Part B: Describe the Graph of the Solution
The graph will represent the intervals where the inequality holds true. We have two intervals on the x-axis:
[tex]\[
(-\infty, -20] \quad \text{and} \quad [36, \infty)
\][/tex]
This means that on the x-axis, the solution set includes all values of [tex]\(x\)[/tex] less than or equal to [tex]\(-20\)[/tex], and all values of [tex]\(x\)[/tex] greater than or equal to [tex]\(36\)[/tex].
On the graph, these intervals will be represented by shading:
- The interval [tex]\((- \infty, -20]\)[/tex] will be shaded, including the point [tex]\(-20\)[/tex].
- The interval [tex]\([36, \infty)\)[/tex] will be shaded, including the point [tex]\(36\)[/tex].
Therefore, the graph of the solution will have shaded intervals at [tex]\((- \infty, -20]\)[/tex] and [tex]\([36, \infty)\)[/tex] on the x-axis.
