Question

Select ALL equations that have one solution.

A.) -3(x - 2) = 3(x - 2)
B.) x + x - (x + x) = 2x - x + 2
C.) 3/4(4x - 8) = 18
D.) 2x + 2x + 2 = 4x + 2
E.) 1/2(x) = x + 1/2
Option 1: A and C
Option 2: B and D
Option 3: A and E
Option 4: C and D

Answer 1

Equations A, B, and C have one solution after simplification, while D has infinite solutions, and E does not have a solution for x at all. However, the correct combination A and C is not listed among the options, thus there is no completely correct answer provided.

Step-by-step explanation:

To determine which equations have one solution, we need to simplify each equation and see if we can solve for x:

1. A.) -3(x - 2) = 3(x - 2): Simplifying both sides we get -3x + 6 = 3x - 6. This simplifies to -6x = -12, which has one solution, x = 2.
2. B.) x + x - (x + x) = 2x - x + 2: Simplifying we get 0 = x + 2. This is also one solution, x = -2.
3. C.) 3/4(4x - 8) = 18: Simplifying we get 3x - 6 = 18. Solving for x gives us one solution, x = 8.
4. D.) 2x + 2x + 2 = 4x + 2: Simplifying we get 4x + 2 = 4x + 2. Subtracting 4x + 2 from both sides we get 0 = 0, which is true for all x, so there are infinite solutions, and hence it does not have only one solution.
5. E.) 1/2(x) = x + 1/2: Simplifying we have 1/2x = x + 1/2. If we subtract 1/2x from both sides, we get 0 = 1/2x + 1/2, which does not have a single solution for x.

Thus, the equations that have one solution are A, B, and C. The correct choice is Option 2: B and D, which incorrectly includes D that has infinite solutions. The right combination should include A and C instead. Therefore, none of the given options is entirely correct.