To determine which expressions are equivalent to the given expression [tex]\(3x - 12 - 2(x + 12)\)[/tex], we need to simplify and compare each expression.
1. Given expression:
[tex]\[
3x - 12 - 2(x + 12)
\][/tex]
Simplify within the parentheses:
[tex]\[
3x - 12 - 2x - 24
\][/tex]
Combine like terms:
[tex]\[
x - 36
\][/tex]
So, we need to find the expressions that simplify to [tex]\(x - 36\)[/tex].
2. Option 1: [tex]\(3x - 12 - 2x + 24\)[/tex]
Simplify by combining like terms:
[tex]\[
3x - 2x - 12 + 24 = x + 12
\][/tex]
This does not match [tex]\(x - 36\)[/tex].
3. Option 2: [tex]\(3(x - 4) - 2x - 24\)[/tex]
Simplify within the parentheses:
[tex]\[
3x - 12 - 2x - 24
\][/tex]
Combine like terms:
[tex]\[
x - 36
\][/tex]
This matches [tex]\(x - 36\)[/tex].
4. Option 3: [tex]\(3(x - 4) - 2(x + 12)\)[/tex]
Simplify within the parentheses:
[tex]\[
3x - 12 - 2x - 24
\][/tex]
Combine like terms:
[tex]\[
x - 36
\][/tex]
This matches [tex]\(x - 36\)[/tex].
5. Option 4: [tex]\(3(x + 4) - 2(x + 12)\)[/tex]
Simplify within the parentheses:
[tex]\[
3x + 12 - 2x - 24
\][/tex]
Combine like terms:
[tex]\[
x - 12
\][/tex]
This does not match [tex]\(x - 36\)[/tex].
6. Option 5: [tex]\(x + 12\)[/tex]
This is already simplified and it does not match [tex]\(x - 36\)[/tex].
7. Option 6: [tex]\(x - 36\)[/tex]
This is already simplified and matches [tex]\(x - 36\)[/tex].
Therefore, the correct expressions that are equivalent to [tex]\(3x - 12 - 2(x + 12)\)[/tex] are:
- [tex]\(3(x - 4) - 2x - 24\)[/tex]
- [tex]\(3(x - 4) - 2(x + 12)\)[/tex]
- [tex]\(x - 36\)[/tex]
These correspond to options [tex]\(2\)[/tex], [tex]\(3\)[/tex], and [tex]\(6\)[/tex].
