To solve the inequality [tex]\(-3(w + 4) + 7 < -23\)[/tex], follow these steps:
1. Distribute the [tex]\(-3\)[/tex] across the terms inside the parentheses:
[tex]\[
-3(w + 4) + 7 < -23
\][/tex]
[tex]\[
-3w - 12 + 7 < -23
\][/tex]
2. Combine like terms:
[tex]\[
-3w - 5 < -23
\][/tex]
3. Isolate [tex]\(w\)[/tex] by first adding 5 to both sides of the inequality:
[tex]\[
-3w - 5 + 5 < -23 + 5
\][/tex]
[tex]\[
-3w < -18
\][/tex]
4. Divide both sides by [tex]\(-3\)[/tex], remembering that dividing by a negative number reverses the inequality sign:
[tex]\[
\frac{-3w}{-3} > \frac{-18}{-3}
\][/tex]
[tex]\[
w > 6
\][/tex]
The solution to the inequality is [tex]\(w > 6\)[/tex].
Correct choice:
D. [tex]\(w >\)[/tex] 6
Graphing the solution:
On a number line, represent [tex]\(w > 6\)[/tex] by drawing an open circle at [tex]\(6\)[/tex] and shading everything to the right of 6.
Here's what the graph looks like:
[tex]\[
\begin{array}{ccccccccccccccc}
\text{(---)} & \text{6)} & \text{(========================)} \\
\end{array}
\][/tex]
The open circle on 6 indicates that 6 is not included in the solution, and the shading to the right illustrates all values greater than 6 are included.
