Sure, let's solve each equation step-by-step and check the solutions.
### Part (a): Solve [tex]\( 4(x-1) - 2(3x + 5) = -3x - 1 \)[/tex]
1. Distribute the coefficients on the left-hand side:
[tex]\[
4(x-1) - 2(3x+5) = 4x - 4 - 6x - 10
\][/tex]
2. Simplify the left-hand side by combining like terms:
[tex]\[
4x - 4 - 6x - 10 = -2x - 14
\][/tex]
3. Write the equation with the simplified left-hand side:
[tex]\[
-2x - 14 = -3x - 1
\][/tex]
4. Add [tex]\(3x\)[/tex] to both sides to start isolating [tex]\(x\)[/tex]:
[tex]\[
-2x + 3x - 14 = -1
\][/tex]
[tex]\[
x - 14 = -1
\][/tex]
5. Add 14 to both sides to completely isolate [tex]\(x\)[/tex]:
[tex]\[
x - 14 + 14 = -1 + 14
\][/tex]
[tex]\[
x = 13
\][/tex]
6. Check the solution by substituting [tex]\(x = 13\)[/tex] back into the original equation:
[tex]\[
4(13-1) - 2(3 \cdot 13 + 5) = -3 \cdot 13 - 1
\][/tex]
[tex]\[
4 \cdot 12 - 2 \cdot 44 = -39 - 1
\][/tex]
[tex]\[
48 - 88 = -40
\][/tex]
[tex]\[
-40 = -40
\][/tex]
The solution [tex]\(x = 13\)[/tex] checks out.
### Part (b): Solve [tex]\( 3x - 5 = 2.5x + 3 - x \)[/tex]
1. Combine like terms on the right-hand side:
[tex]\[
2.5x + 3 - x = 1.5x + 3
\][/tex]
2. Write the equation with the simplified right-hand side:
[tex]\[
3x - 5 = 1.5x + 3
\][/tex]
3. Subtract [tex]\(1.5x\)[/tex] from both sides to start isolating [tex]\(x\)[/tex]:
[tex]\[
3x - 1.5x - 5 = 3
\][/tex]
[tex]\[
1.5x - 5 = 3
\][/tex]
4. Add 5 to both sides to isolate [tex]\(1.5x\)[/tex]:
[tex]\[
1.5x - 5 + 5 = 3 + 5
\][/tex]
[tex]\[
1.5x = 8
\][/tex]
5. Divide both sides by [tex]\(1.5\)[/tex] to solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{8}{1.5}
\][/tex]
[tex]\[
x \approx 5.3333
\][/tex]
6. Check the solution by substituting [tex]\(x = 5.3333\)[/tex] back into the original equation:
[tex]\[
3(5.3333) - 5 = 2.5(5.3333) + 3 - (5.3333)
\][/tex]
[tex]\[
15.9999 - 5 = 13.3333 + 3 - 5.3333
\][/tex]
[tex]\[
10.9999 \approx 10.9999
\][/tex]
The solution [tex]\(x \approx 5.3333\)[/tex] checks out.
### Final Answers:
- Part (a): [tex]\( x = 13 \)[/tex]
- Part (b): [tex]\( x \approx 5.3333 \)[/tex]
