Let's examine each of the equations to determine which, if any, have no solution.
### Equation 1:
[tex]\[ x - 5(x-4) = 2(2x+7) + 6 \][/tex]
1. Expand and simplify both sides:
[tex]\[ x - 5x + 20 = 4x + 14 + 6 \][/tex]
[tex]\[ -4x + 20 = 4x + 20 \][/tex]
2. Move all terms to one side:
[tex]\[ -4x - 4x + 20 - 20 = 0 \][/tex]
[tex]\[ -8x = 0 \][/tex]
3. Solve for [tex]\( x \)[/tex]:
[tex]\[ x = 0 \][/tex]
This equation has a solution: [tex]\( x = 0 \)[/tex].
### Equation 2:
[tex]\[ 5(x + 1) - 3x = 5 - 2(5 - x) \][/tex]
1. Expand and simplify both sides:
[tex]\[ 5x + 5 - 3x = 5 - 10 + 2x \][/tex]
[tex]\[ 2x + 5 = 5 - 10 + 2x \][/tex]
2. Combine like terms:
[tex]\[ 2x + 5 = -5 + 2x \][/tex]
3. Subtract [tex]\( 2x \)[/tex] from both sides:
[tex]\[ 5 = -5 \][/tex]
The left side does not equal the right side, so there is no solution for this equation.
### Equation 3:
[tex]\[ 1 - 3(x + 3) = 2(3x - 4) - 9x \][/tex]
1. Expand and simplify both sides:
[tex]\[ 1 - 3x - 9 = 6x - 8 - 9x \][/tex]
[tex]\[ -3x - 8 = -3x - 8 \][/tex]
2. Combine like terms:
[tex]\[ -3x - 8 = -3x - 8 \][/tex]
This equation is an identity; it is true for all [tex]\( x \)[/tex], hence it has infinitely many solutions.
### Equation 4:
[tex]\[ 3(x - 4) + x = 4(3 - x) \][/tex]
1. Expand and simplify both sides:
[tex]\[ 3x - 12 + x = 12 - 4x \][/tex]
[tex]\[ 4x - 12 = 12 - 4x \][/tex]
2. Move all terms to one side:
[tex]\[ 4x + 4x = 12 + 12 \][/tex]
[tex]\[ 8x = 24 \][/tex]
3. Solve for [tex]\( x \)[/tex]:
[tex]\[ x = 3 \][/tex]
This equation has a solution: [tex]\( x = 3 \)[/tex].
### Conclusion:
From our analysis:
- Equation 1 has a solution: [tex]\( x = 0 \)[/tex].
- Equation 2 has no solution.
- Equation 3 has infinitely many solutions.
- Equation 4 has a solution: [tex]\( x = 3 \)[/tex].
Therefore, the equation that has no solution is:
[tex]\[ 5(x + 1) - 3x = 5 - 2(5 - x) \][/tex]
