To solve the equation [tex]\(-4(3 + x) + 5 = 4(x + 3)\)[/tex], we'll go through the steps methodically.
1. Distribute the constants inside the parentheses:
- On the left side of the equation:
[tex]\(-4(3 + x)\)[/tex] expands to [tex]\(-4 \cdot 3 - 4 \cdot x = -12 - 4x\)[/tex].
So, the left side becomes: [tex]\(-12 - 4x + 5\)[/tex].
- On the right side of the equation:
[tex]\(4(x + 3)\)[/tex] expands to [tex]\(4 \cdot x + 4 \cdot 3 = 4x + 12\)[/tex].
So, the right side remains: [tex]\(4x + 12\)[/tex].
2. Simplify the left side of the equation:
Combine the constant terms on the left:
[tex]\[
-12 - 4x + 5 = -7 - 4x
\][/tex]
3. Rewrite the equation with the simplified left side:
[tex]\[
-7 - 4x = 4x + 12
\][/tex]
4. Combine like terms by moving all [tex]\(x\)[/tex] terms to one side and all the constant terms to the other side:
- Add [tex]\(4x\)[/tex] to both sides to move the [tex]\(x\)[/tex] terms to one side:
[tex]\[
-7 - 4x + 4x = 4x + 12 + 4x
\][/tex]
This simplifies to:
[tex]\[
-7 = 8x + 12
\][/tex]
- Next, move the constant term from the right side to the left side by subtracting 12 from both sides:
[tex]\[
-7 - 12 = 8x + 12 - 12
\][/tex]
This simplifies to:
[tex]\[
-19 = 8x
\][/tex]
5. Solve for [tex]\(x\)[/tex]:
- Divide both sides by 8 to isolate [tex]\(x\)[/tex]:
[tex]\[
x = \frac{-19}{8}
\][/tex]
Simplifying the fraction, we get:
[tex]\[
x = -2.375
\][/tex]
Therefore, the solution to the equation is:
[tex]\[
x = -2.375
\][/tex]
