To determine how many solutions exist for the given equation:
[tex]\[ 3(x-2) = 22 - x \][/tex]
Let's solve the equation step-by-step:
1. Distribute the 3 on the left side:
[tex]\[ 3 \cdot x - 3 \cdot 2 = 22 - x \][/tex]
Which simplifies to:
[tex]\[ 3x - 6 = 22 - x \][/tex]
2. Move all terms involving [tex]\(x\)[/tex] to one side of the equation:
We add [tex]\(x\)[/tex] to both sides to get the [tex]\(x\)[/tex] terms together:
[tex]\[ 3x + x - 6 = 22 \][/tex]
This simplifies to:
[tex]\[ 4x - 6 = 22 \][/tex]
3. Isolate the [tex]\(x\)[/tex] term:
Add 6 to both sides to move the constant to the right side:
[tex]\[ 4x - 6 + 6 = 22 + 6 \][/tex]
This simplifies to:
[tex]\[ 4x = 28 \][/tex]
4. Solve for [tex]\(x\)[/tex]:
Divide by 4:
[tex]\[ \frac{4x}{4} = \frac{28}{4} \][/tex]
This simplifies to:
[tex]\[ x = 7 \][/tex]
After all these steps, we have found [tex]\(x = 7\)[/tex].
Conclusively, the equation [tex]\(3(x-2)=22-x\)[/tex] has a single unique solution. Therefore, the number of solutions is:
One
