Sure, let's solve the compound inequality step by step:
### Step 1: Solve the inequality [tex]\( x + 5 \leq 1 \)[/tex]
Subtract 5 from both sides:
[tex]\[
x + 5 - 5 \leq 1 - 5
\][/tex]
[tex]\[
x \leq -4
\][/tex]
### Step 2: Solve the inequality [tex]\( x - 7 \geq -3 \)[/tex]
Add 7 to both sides:
[tex]\[
x - 7 + 7 \geq -3 + 7
\][/tex]
[tex]\[
x \geq 4
\][/tex]
### Step 3: Combine the solutions
The compound inequality is an "or" statement:
[tex]\[
x + 5 \leq 1 \text{ or } x - 7 \geq -3
\][/tex]
Combining the solutions, we get:
[tex]\[
x \leq -4 \text{ or } x \geq 4
\][/tex]
### Step 4: Identify the graph of the solution
- For [tex]\( x \leq -4 \)[/tex], it includes all values to the left of -4 including -4.
- For [tex]\( x \geq 4 \)[/tex], it includes all values to the right of 4 including 4.
Thus, the correct answer is:
A. [tex]\(x \leq -4 \)[/tex] or [tex]\( x \geq 4 \)[/tex]
### Graph of the Solution
On a number line, this would be represented as two segments:
- A shaded circle at -4 extending indefinitely to the left.
- A shaded circle at 4 extending indefinitely to the right.
