Sure, let's solve the equation step-by-step to determine which option is equivalent to the given equation:
Given equation:
[tex]\[ 3(x + 4) = -(2 - x) \][/tex]
First, we simplify both sides of the equation.
1. Expand the left side:
[tex]\[ 3(x + 4) = 3x + 12 \][/tex]
2. Expand the right side:
[tex]\[ -(2 - x) = -2 + x \][/tex]
Now, the equation becomes:
[tex]\[ 3x + 12 = -2 + x \][/tex]
Next, we simplify the equation further by getting all the [tex]\(x\)[/tex] terms on one side and the constants on the other side.
3. Subtract [tex]\(x\)[/tex] from both sides:
[tex]\[ 3x + 12 - x = -2 + x - x \][/tex]
[tex]\[ 2x + 12 = -2 \][/tex]
4. Subtract 12 from both sides:
[tex]\[ 2x + 12 - 12 = -2 - 12 \][/tex]
[tex]\[ 2x = -14 \][/tex]
5. Divide both sides by 2:
[tex]\[ \frac{2x}{2} = \frac{-14}{2} \][/tex]
[tex]\[ x = -7 \][/tex]
Now, let's check each option to see which one simplifies to the equivalent expression:
A) [tex]\(3x + 12 = -2 + x\)[/tex]
[tex]\[ 3x + 12 - x = -2 + x - x \][/tex]
[tex]\[ 2x + 12 = -2 \][/tex]
[tex]\[ x = -7 \][/tex]
This is not equivalent.
B) [tex]\(3x + 4 = -2x + x\)[/tex]
[tex]\[ 3x + 4 = -x \][/tex]
[tex]\[ 3x + x + 4 = -x + x \][/tex]
[tex]\[ 3x + 4 = 0 \][/tex]
[tex]\[ x = -\frac{4}{3} \][/tex]
This is not equivalent.
C) [tex]\(3x - 12 = -2 - x\)[/tex]
[tex]\[ 3x - 12 + x = -2 - x + x \][/tex]
[tex]\[ 4x - 12 = -2 \][/tex]
[tex]\[ x - 12 = -2 \][/tex]
[tex]\[ x = 10 \][/tex]
This is not equivalent.
D) The provided option does not make sense algebraically.
From the simplified version of the original equation, we conclude that none of the options [tex]\(A\)[/tex], [tex]\(B\)[/tex], [tex]\(C\)[/tex], or [tex]\(D\)[/tex] are equivalent to the given problem statement. This matches our previous results, indicating an issue with the provided options themselves.
