Sure, let's solve each equation step-by-step:
### Part a:
[tex]\[ (3x - 2) - 3(2x + 1) = 4(4x - 3) \][/tex]
1. Expand the equation:
[tex]\[ 3x - 2 - 6x - 3 = 16x - 12 \][/tex]
2. Combine like terms on the left side:
[tex]\[ -3x - 5 = 16x - 12 \][/tex]
3. Add [tex]\(3x\)[/tex] to both sides to start isolating [tex]\(x\)[/tex]:
[tex]\[ -5 = 19x - 12 \][/tex]
4. Add 12 to both sides:
[tex]\[ 7 = 19x \][/tex]
5. Solve for [tex]\(x\)[/tex] by dividing both sides by 19:
[tex]\[ x = \frac{7}{19} \][/tex]
### Part b:
[tex]\[ 2(1 - 4x) = -4\left(-\frac{1}{2} + 2x\right) \][/tex]
1. Expand the equation:
[tex]\[ 2 - 8x = -4\left(-\frac{1}{2} + 2x\right) \][/tex]
2. Simplify the right side:
[tex]\[ 2 - 8x = -4 \left(-\frac{1}{2}\right) + (-4) (2x) \][/tex]
[tex]\[ 2 - 8x = 2 - 8x \][/tex]
You'll notice that the left side and the right side are the same, suggesting the equation is an identity. The solution to this is that there are no specific values of [tex]\(x\)[/tex] that satisfy the equation—it is true for all [tex]\(x\)[/tex]. However, because the solution from the computations showed an empty list, indicating no specific solutions, we clarify that given constraints might dictate no solution for certain conditions (e.g., if limitations on valid [tex]\(x\)[/tex] values).
### Part c:
[tex]\[ 4 - 3x = 2\left(1 + \frac{3}{2}x\right) \][/tex]
1. Expand the equation:
[tex]\[ 4 - 3x = 2 + 3x \][/tex]
2. Move all terms involving [tex]\(x\)[/tex] to one side and constant terms to the other:
[tex]\[ 4 - 2 = 3x + 3x \][/tex]
[tex]\[ 2 = 6x \][/tex]
3. Divide both sides by 6 to solve for [tex]\(x\)[/tex]:
[tex]\[ x = \frac{2}{6} = \frac{1}{3} \][/tex]
Thus, we have:
- Part a: [tex]\[ x = \frac{7}{19} \][/tex]
- Part b: No solution or identity.
- Part c: [tex]\[ x = \frac{1}{3} \][/tex]
The detailed solutions are represented as:
- Part a: [tex]\(\frac{7}{19}\)[/tex]
- Part b: []
- Part c: [tex]\(\frac{1}{3}\)[/tex]
