To determine which equation has no solution, we analyze each given equation individually. Here is a detailed step-by-step exploration for each equation:
### Equation 1
[tex]\( -\frac{1}{2}(8x + 6) - 2x = -3(2x + 1) \)[/tex]
Simplify both sides:
[tex]\[
-\frac{1}{2}(8x + 6) - 2x = -3(2x + 1)
\][/tex]
[tex]\[
-4x - 3 - 2x = -6x - 3
\][/tex]
[tex]\[
-6x - 3 = -6x - 3
\][/tex]
This simplifies to:
[tex]\[
-6x - 3 = -6x - 3
\][/tex]
Both sides are identically equal, suggesting that this equation has infinitely many solutions for any value of [tex]\( x \)[/tex].
### Equation 2
[tex]\( -4\left(\frac{1}{2}x + 2\right) = -2x - 8 + 4x \)[/tex]
Simplify both sides:
[tex]\[
-4\left(\frac{1}{2}x + 2\right) = -2x - 8 + 4x
\][/tex]
[tex]\[
-2x - 8 = 2x - 8
\][/tex]
[tex]\[
-2x - 8 = 2x - 8
\][/tex]
Rearrange the equation:
[tex]\[
-2x - 2x = -8 + 8
\][/tex]
[tex]\[
-4x = 0
\][/tex]
[tex]\[
x = 0
\][/tex]
This equation has one solution [tex]\( x = 0 \)[/tex].
### Equation 3
[tex]\( 7(x + 2) - 3x = \frac{2}{3}(6x + 3) \)[/tex]
Simplify both sides:
[tex]\[
7(x + 2) - 3x = \frac{2}{3}(6x + 3)
\][/tex]
[tex]\[
7x + 14 - 3x = 4x + 2
\][/tex]
[tex]\[
4x + 14 = 4x + 2
\][/tex]
Rearrange the equation:
[tex]\[
4x - 4x + 14 = 2
\][/tex]
[tex]\[
14 \neq 2
\][/tex]
This produces a contradiction, which means the equation has no solution.
### Equation 4
[tex]\( -2x + 5 - 3x + 12 = -5(x - 3) + 2 \)[/tex]
Simplify both sides:
[tex]\[
-2x + 5 - 3x + 12 = -5(x - 3) + 2
\][/tex]
[tex]\[
-5x + 17 = -5x + 15 + 2
\][/tex]
[tex]\[
-5x + 17 = -5x + 17
\][/tex]
This simplifies to:
[tex]\[
-5x + 17 = -5x + 17
\][/tex]
Both sides are identically equal, suggesting that this equation has infinitely many solutions for any value of [tex]\( x \)[/tex].
### Conclusion
Thus, the equation that has no solution is:
[tex]\[ 7(x + 2) - 3x = \frac{2}{3}(6x + 3) \][/tex]
So, the correct answer is:
[tex]\[ 7(x + 2) - 3x = \frac{2}{3}(6x + 3) \][/tex]
