Sure, let's solve each equation step-by-step.
### Part (a):
Solve the equation [tex]\(5(2+w) - w = 10 + 4(w+1)\)[/tex]
1. Distribute the numbers inside the parentheses:
[tex]\[
5(2+w) = 5 \cdot 2 + 5 \cdot w = 10 + 5w
\][/tex]
[tex]\[
4(w+1) = 4 \cdot w + 4 \cdot 1 = 4w + 4
\][/tex]
2. Rewrite the equation:
[tex]\[
10 + 5w - w = 10 + 4w + 4
\][/tex]
3. Combine like terms:
[tex]\[
10 + 4w = 14 + 4w
\][/tex]
4. Move all variables to one side and constants to the other:
Subtract [tex]\(4w\)[/tex] and 14 from both sides:
[tex]\[
10 + 4w - 4w = 14 + 4w - 4w
\][/tex]
[tex]\[
10 = 14
\][/tex]
Since [tex]\(10\)[/tex] is not equal to [tex]\(14\)[/tex], this means there are no solutions for this equation.
### Part (b):
Solve the equation [tex]\(2(x-1) + 8 = 6(2x-4)\)[/tex]
1. Distribute the numbers inside the parentheses:
[tex]\[
2(x-1) = 2 \cdot x - 2 \cdot 1 = 2x - 2
\][/tex]
[tex]\[
6(2x-4) = 6 \cdot 2x - 6 \cdot 4 = 12x - 24
\][/tex]
2. Rewrite the equation:
[tex]\[
2x - 2 + 8 = 12x - 24
\][/tex]
3. Combine like terms:
[tex]\[
2x + 6 = 12x - 24
\][/tex]
4. Move all variables to one side and constants to the other:
Subtract [tex]\(12x\)[/tex] and 6 from both sides:
[tex]\[
2x - 12x + 6 = -24
\][/tex]
[tex]\[
-10x + 6 = -24
\][/tex]
Subtract 6 from both sides:
[tex]\[
-10x = -30
\][/tex]
5. Solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{-30}{-10} = 3
\][/tex]
So, the solution for this equation is [tex]\(x = 3\)[/tex].
### Summary
(a) There are no solutions for the equation [tex]\(5(2+w) - w = 10 + 4(w+1)\)[/tex].
(b) The solution for the equation [tex]\(2(x-1) + 8 = 6(2x-4)\)[/tex] is [tex]\(x = 3\)[/tex].
