When an equation has no solution, it means that no x can make the equation be a true statement. The opposite is when an equation has infinitely many solutions, it means everything put inf or x makes it true. One solution lies in between, and there is exactly one x that makes it true.
Let's look at the equations and see which are which.
A--
3 (x + 4) = 3x + 11
3x + 12 = 3x + 11 by the distributive property on the left side
12 = 11 by subtracting 3x on both sides.
Notice here that 12 = 11. That's always false. This kind of always false statement, rather than x = _____, tells us we have no solutions.
B---
-2 (x + 3) = -2x - 6
-2x - 6 = -2x -6
-6 = -6 by similar steps as in A
This equation gives something that is always true, that -6 = -6. When you get the same thing on both sides, this is a clue that you have infinitely many solutions.
C---
4 (x - 1) = 1/2 (x - 8)
4x - 4 = 1/2x - 4 by distributing on both sides
8x - 8 = x - 8 by multiplying the whole equation by 2 to remove fractions.
7x - 8 = -8 by subtracting x from both sides
7x = 0 by adding 8 to both sides
x = 0 by dividing 7 on both sides.
Here, we have exactly one solution, zero.
D---
This equation should be checked given the presence of a double equals sign.
Thus, we have the following.
A - No solution
B - Infinitely many solutions
C - Exactly one solution at x = 0
D - Check the double equals to see if it's a typo.
