Of course! Let's go through the steps for each equation to separate the variable and solve.
### Part (a) [tex]\( 2(x - 1) = 10 \)[/tex]
1. Distribute the 2 on the left side:
[tex]\[
2(x - 1) = 10 \implies 2x - 2 = 10
\][/tex]
2. Isolate the term with the variable (get 2x by itself):
[tex]\[
2x - 2 = 10 \implies 2x = 10 + 2 \implies 2x = 12
\][/tex]
3. Solve for [tex]\( x \)[/tex] by dividing both sides by 2:
[tex]\[
x = \frac{12}{2} \implies x = 6
\][/tex]
### Part (b) [tex]\( \frac{x}{2} + \frac{1}{2} = 1 \)[/tex]
1. Isolate the term with the variable (subtract [tex]\(\frac{1}{2}\)[/tex] from both sides):
[tex]\[
\frac{x}{2} + \frac{1}{2} = 1 \implies \frac{x}{2} = 1 - \frac{1}{2} \implies \frac{x}{2} = \frac{1}{2}
\][/tex]
2. Solve for [tex]\( x \)[/tex] by multiplying both sides by 2:
[tex]\[
x = \left(\frac{1}{2}\right) \times 2 \implies x = 1
\][/tex]
### Part (c) [tex]\( -2 - x = 6 \)[/tex]
1. Isolate the term with the variable (add 2 to both sides):
[tex]\[
-2 - x = 6 \implies -x = 6 + 2 \implies -x = 8
\][/tex]
2. Solve for [tex]\( x \)[/tex] by multiplying both sides by -1:
[tex]\[
x = -8
\][/tex]
### Summary of Solutions
- Part (a) [tex]\( x = 6 \)[/tex]
- Part (b) [tex]\( x = 1 \)[/tex]
- Part (c) [tex]\( x = -8 \)[/tex]
These are the steps to isolate the variable and solve for [tex]\( x \)[/tex] in each of the given equations.
