To solve the compound inequality [tex]\(-18 > -5x + 2 \geq -48\)[/tex], we need to break it down into two parts: [tex]\(-18 > -5x + 2\)[/tex] and [tex]\(-5x + 2 \geq -48\)[/tex].
1. Solve [tex]\(-18 > -5x + 2\)[/tex]:
First, isolate [tex]\(x\)[/tex] by performing the appropriate operations on both sides of the inequality:
[tex]\[
-18 > -5x + 2
\][/tex]
Subtract 2 from both sides:
[tex]\[
-18 - 2 > -5x
\][/tex]
[tex]\[
-20 > -5x
\][/tex]
To solve for [tex]\(x\)[/tex], divide both sides by -5. Remember to reverse the inequality when dividing by a negative number:
[tex]\[
\frac{-20}{-5} < x
\][/tex]
[tex]\[
4 < x
\][/tex]
or
[tex]\[
x > 4
\][/tex]
2. Solve [tex]\(-5x + 2 \geq -48\)[/tex]:
Isolate [tex]\(x\)[/tex] similarly:
[tex]\[
-5x + 2 \geq -48
\][/tex]
Subtract 2 from both sides:
[tex]\[
-5x \geq -50
\][/tex]
Divide by -5, reversing the inequality:
[tex]\[
x \leq \frac{-50}{-5}
\][/tex]
[tex]\[
x \leq 10
\][/tex]
3. Combine the two inequalities:
From the solutions of each part:
[tex]\[
x > 4
\][/tex]
and
[tex]\[
x \leq 10
\][/tex]
Combine the solutions to form the compound inequality:
[tex]\[
4 < x \leq 10
\][/tex]
4. Represent on a number line:
The correct graph for the solution [tex]\(4 < x \leq 10\)[/tex] will show the portion of the number line between (but not including) 4 and (including) 10. This can be seen as an open circle at 4 and a closed circle at 10 with a line connecting them.
[tex]\[
\begin{array}{lllllllllllllllllll}
-5 & -4 & -3 & -2 & -1 & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 \\
& & & & & & & & & \circ & ----- & ----- & ----- & ----- & ----- & \bullet & \\
\end{array}
\][/tex]
