Certainly! Let's solve the equation [tex]\(8(x+2) = -32\)[/tex] step-by-step and choose the appropriate reasons for each step.
### Step-by-Step Solution
1. Given equation:
[tex]\[
8(x + 2) = -32
\][/tex]
Reason: This is the given equation.
2. Apply the Distributive Property:
[tex]\[
8x + 16 = -32
\][/tex]
Reason: Distributive Property. The distributive property is used to multiply 8 with both [tex]\(x\)[/tex] and 2.
3. Subtract 16 from both sides:
[tex]\[
8x + 16 - 16 = -32 - 16
\][/tex]
Reason: Subtract 16 from both sides. This step is done to isolate the term containing [tex]\(x\)[/tex] on one side of the equation.
4. Simplify both sides:
[tex]\[
8x = -48
\][/tex]
Reason: Simplify both sides. Simplifying [tex]\(+16 - 16\)[/tex] on the left-hand side and [tex]\(-32 - 16\)[/tex] on the right-hand side.
5. Divide both sides by 8:
[tex]\[
\frac{8x}{8} = \frac{-48}{8}
\][/tex]
Reason: Divide both sides by 8. This step is done to solve for [tex]\(x\)[/tex] by getting rid of the coefficient 8.
6. Simplify both sides after division:
[tex]\[
x = -6
\][/tex]
Reason: Simplify both sides after division. The division simplifies to [tex]\(x = -6\)[/tex].
### Filled-in Table
\begin{tabular}{|c|l|}
\hline
Step & Reason \\
\hline
[tex]$8(x+2) = -32$[/tex] & Given equation \\
\hline
[tex]$8x + 16 = -32$[/tex] & Distributive Property \\
\hline
[tex]$8x + 16 - 16 = -32 - 16$[/tex] & Subtract 16 from both sides \\
\hline
[tex]$8x = -48$[/tex] & Simplify both sides \\
\hline
[tex]$\frac{8x}{8} = \frac{-48}{8}$[/tex] & Divide both sides by 8 \\
\hline
[tex]$x = -6$[/tex] & Simplify both sides after division \\
\hline
\end{tabular}
This detailed solution walks you through each step clearly, ensuring that you understand the rationale for each operation performed.
