Let's solve the equation [tex]\(\frac{-22 + 3x}{3x + 7} = 2\)[/tex] step by step.
1. Multiply both sides of the equation by the denominator [tex]\((3x + 7)\)[/tex]:
[tex]\[
\frac{-22 + 3x}{3x + 7} \cdot (3x + 7) = 2 \cdot (3x + 7)
\][/tex]
This simplifies to:
[tex]\[
-22 + 3x = 2(3x + 7)
\][/tex]
2. Distribute the right-hand side:
[tex]\[
-22 + 3x = 2 \cdot 3x + 2 \cdot 7
\][/tex]
Which gives:
[tex]\[
-22 + 3x = 6x + 14
\][/tex]
3. Isolate the variable [tex]\(x\)[/tex] by moving all terms involving [tex]\(x\)[/tex] to one side and constant terms to the other side:
Subtract [tex]\(3x\)[/tex] from both sides:
[tex]\[
-22 + 3x - 3x = 6x + 14 - 3x
\][/tex]
This simplifies to:
[tex]\[
-22 = 3x + 14
\][/tex]
4. Move the constant term from the right to the left side:
Subtract 14 from both sides:
[tex]\[
-22 - 14 = 3x + 14 - 14
\][/tex]
This simplifies to:
[tex]\[
-36 = 3x
\][/tex]
5. Solve for [tex]\(x\)[/tex] by dividing both sides by 3:
[tex]\[
\frac{-36}{3} = \frac{3x}{3}
\][/tex]
This gives:
[tex]\[
x = -12
\][/tex]
Therefore, the solution of the equation is [tex]\(\boxed{-12}\)[/tex].
