To solve the given compound inequality [tex]\(-18 > -5x + 2 \geq -48\)[/tex], let's break it down into two separate inequalities and solve them step-by-step.
### Solving the First Inequality:
[tex]\[
-18 > -5x + 2
\][/tex]
1. Subtract 2 from both sides:
[tex]\[
-18 - 2 > -5x
\][/tex]
2. Simplify the left-hand side:
[tex]\[
-20 > -5x
\][/tex]
3. Divide both sides by [tex]\(-5\)[/tex]. Remember, dividing by a negative number reverses the inequality sign:
[tex]\[
4 < x \quad \text{or} \quad x > 4
\][/tex]
### Solving the Second Inequality:
[tex]\[
-5x + 2 \geq -48
\][/tex]
1. Subtract 2 from both sides:
[tex]\[
-5x \geq -50
\][/tex]
2. Divide both sides by [tex]\(-5\)[/tex]. Again, dividing by a negative number reverses the inequality sign:
[tex]\[
x \leq 10
\][/tex]
### Combining the Results:
From solving the two inequalities, we have:
[tex]\[
4 < x \quad \text{and} \quad x \leq 10
\][/tex]
We can combine these to express the solution as:
[tex]\[
4 < x \leq 10
\][/tex]
### Graphing the Solution:
On a number line, this solution indicates that [tex]\(x\)[/tex] is greater than 4 but less than or equal to 10. Here's how we represent this graphically:
- The point at [tex]\(4\)[/tex] is represented by an open circle (indicating that [tex]\(4\)[/tex] is not included in the solution).
- The interval from just above [tex]\(4\)[/tex] up to and including [tex]\(10\)[/tex] is shaded.
- The point at [tex]\(10\)[/tex] is represented by a closed circle (indicating that [tex]\(10\)[/tex] is included in the solution).
Below is the graphical representation:
[tex]\[
\begin{array}{lllllllllllllllllll}
-5 & -4 & -3 & -2 & -1 & 0 & 1 & 2 & 3 & \circ 4 & 5 & 6 & 7 & 8 & 9 & \bullet 10 & 11 & 12 & 13 \\
& & & & & & & & & \text{------------------------} & & & & & & & & & &
\end{array}
\][/tex]
The shaded line indicates that all [tex]\(x\)[/tex] values greater than [tex]\(4\)[/tex] and up to [tex]\(10\)[/tex] are part of the solution, including [tex]\(10\)[/tex].
