To solve the inequality [tex]\(-8 \geq -5x + 2 > -38\)[/tex], we will break it into two separate inequalities and solve them step-by-step.
### Step 1: Solve [tex]\(-8 \geq -5x + 2\)[/tex]
1. Subtract 2 from both sides:
[tex]\[
-8 - 2 \geq -5x
\][/tex]
[tex]\[
-10 \geq -5x
\][/tex]
2. Divide by -5 (note that dividing by a negative number reverses the inequality sign):
[tex]\[
\frac{-10}{-5} \leq x
\][/tex]
[tex]\[
2 \leq x
\][/tex]
### Step 2: Solve [tex]\(-5x + 2 > -38\)[/tex]
1. Subtract 2 from both sides:
[tex]\[
-5x + 2 - 2 > -38 - 2
\][/tex]
[tex]\[
-5x > -40
\][/tex]
2. Divide by -5 (remember to reverse the inequality sign):
[tex]\[
\frac{-40}{-5} < x
\][/tex]
[tex]\[
8 < x
\][/tex]
### Combine the Results
From Step 1, we have:
[tex]\[
x \geq 2
\][/tex]
From Step 2, we have:
[tex]\[
x > 8
\][/tex]
### Getting the Combined Solution
To satisfy both inequalities, we need to consider the tighter bounds. So, the combined inequality is:
[tex]\[
x > 8
\][/tex]
### Graphing the Solution
To graph the solution [tex]\(x > 8\)[/tex]:
1. Draw a number line.
2. Put an open circle on 8 (to indicate that 8 is not included).
3. Shade the number line to the right of 8 to represent all numbers greater than 8.
The graph looks like this:
[tex]\[ \begin{array}{cccccccc}
& & & & & \circ & \rightarrow & \\
\end{array}
\][/tex]
This open circle at 8 and the shading to the right indicates [tex]\(x > 8\)[/tex].
