Given the equation [tex]\(|3x - 9| - 24 = -12\)[/tex], let's simplify it to understand which relationships are equivalent.
Step-by-Step Simplification:
1. Starting Equation:
[tex]\[
|3x - 9| - 24 = -12
\][/tex]
2. Isolate the Absolute Value Expression:
Add 24 to both sides:
[tex]\[
|3x - 9| - 24 + 24 = -12 + 24
\][/tex]
[tex]\[
|3x - 9| = 12
\][/tex]
So, the equation simplifies to:
[tex]\[
|3x - 9| = 12
\][/tex]
3. Solving the Absolute Value Equation:
The absolute value equation [tex]\(|A| = B\)[/tex] has two solutions:
[tex]\[
A = B \quad \text{or} \quad A = -B
\][/tex]
For our equation [tex]\(|3x - 9| = 12\)[/tex], the solutions are:
[tex]\[
3x - 9 = 12 \quad \text{or} \quad 3x - 9 = -12
\][/tex]
4. Solve for [tex]\(x\)[/tex] in Both Cases:
- Case 1: [tex]\(3x - 9 = 12\)[/tex]
[tex]\[
3x = 12 + 9
\][/tex]
[tex]\[
3x = 21
\][/tex]
[tex]\[
x = 7
\][/tex]
- Case 2: [tex]\(3x - 9 = -12\)[/tex]
[tex]\[
3x = -12 + 9
\][/tex]
[tex]\[
3x = -3
\][/tex]
[tex]\[
x = -1
\][/tex]
So, the solutions to the simplified equation are [tex]\(x = 7\)[/tex] and [tex]\(x = -1\)[/tex].
Based on this, we can determine which relationships are equivalent to [tex]\(|3x - 9| - 24 = -12\)[/tex]:
1. [tex]\(3x - 9 = -36\)[/tex] or [tex]\(3x - 9 = 36\)[/tex]
- These equations are not equivalent to [tex]\(|3x - 9| = 12\)[/tex].
2. [tex]\(x = -9\)[/tex] or [tex]\(x = 15\)[/tex]
- These solutions do not match our simplified equation [tex]\(|3x - 9| = 12\)[/tex].
3. [tex]\(3x - 9 = -12\)[/tex] or [tex]\(3x - 9 = 12\)[/tex]
- These are equivalent to the simplified equation [tex]\(|3x - 9| = 12\)[/tex].
4. [tex]\(x = -1\)[/tex] or [tex]\(x = 7\)[/tex]
- These are the solutions to our simplified equation, so they are equivalent.
5. [tex]\(|3x - 9| = 12\)[/tex]
- This is the simplified form of our original equation, so it is obviously equivalent.
6. [tex]\(|3x - 9| = 36\)[/tex]
- This equation is not equivalent to the simplified form [tex]\(|3x - 9| = 12\)[/tex].
Thus, the correct answers are:
- [tex]\(3 x-9=-12\)[/tex] or [tex]\(3 x-9=12\)[/tex]
- [tex]\(x=-1\)[/tex] or [tex]\(x=7\)[/tex]
- [tex]\(|3 x-9|=12\)[/tex]
