To solve the inequality [tex]\(-8 \geq -5x + 2 > -38\)[/tex], we need to break it down into two separate inequalities and solve each one step-by-step.
### Step-by-Step Solution
#### Step 1: Solve the Left Inequality [tex]\(-8 \geq -5x + 2\)[/tex]
1. Start with the inequality: [tex]\(-8 \geq -5x + 2\)[/tex].
2. Subtract 2 from both sides to isolate the term containing [tex]\(x\)[/tex]:
[tex]\[
-8 - 2 \geq -5x
\][/tex]
[tex]\[
-10 \geq -5x
\][/tex]
3. Divide both sides by [tex]\(-5\)[/tex], remembering to reverse the inequality symbol because we are dividing by a negative number:
[tex]\[
\frac{-10}{-5} \leq x
\][/tex]
[tex]\[
2 \leq x
\][/tex]
So, from the left inequality, we get [tex]\(x \geq 2\)[/tex].
#### Step 2: Solve the Right Inequality [tex]\(-5x + 2 > -38\)[/tex]
1. Start with the inequality: [tex]\(-5x + 2 > -38\)[/tex].
2. Subtract 2 from both sides to isolate the term containing [tex]\(x\)[/tex]:
[tex]\[
-5x > -38 - 2
\][/tex]
[tex]\[
-5x > -40
\][/tex]
3. Divide both sides by [tex]\(-5\)[/tex], remembering to reverse the inequality symbol because we are dividing by a negative number:
[tex]\[
\frac{-40}{-5} < x
\][/tex]
[tex]\[
8 < x
\][/tex]
So, from the right inequality, we get [tex]\(x < 8\)[/tex].
### Combining the Solutions
Now, we combine both parts of the solution:
[tex]\[
2 \leq x < 8
\][/tex]
This means that [tex]\(x\)[/tex] is in the interval [tex]\([2, 8)\)[/tex]. This interval includes all real numbers [tex]\(x\)[/tex] such that [tex]\(x \geq 2\)[/tex] and [tex]\(x < 8\)[/tex].
### Graph of the Solution
The correct graph of the solution to the inequality [tex]\(-8 \geq -5x + 2 > -38\)[/tex] should look like this on a number line:
- There will be a closed (solid) circle at [tex]\(x = 2\)[/tex], indicating that [tex]\(2\)[/tex] is included in the solution.
- There will be an open circle at [tex]\(x = 8\)[/tex], indicating that [tex]\(8\)[/tex] is not included in the solution.
- The line segment between [tex]\(2\)[/tex] and [tex]\(8\)[/tex] will be shaded, representing all values of [tex]\(x\)[/tex] between but not including [tex]\(8\)[/tex].
Here is the graphical representation of the solution:
[tex]\[ \texttt{-|--|--|--|---|---|--( } \][/tex]
This depicts the range of [tex]\(x\)[/tex] values that satisfy [tex]\(2 \leq x < 8\)[/tex].
