Equation (b) 5 + 2(3 + 2x) = x + 3(x + 1) has no solution. When simplified, it results in a false statement, indicating an inconsistent system. The other equations either have infinite solutions or are always true.
To determine which equation has no solution, we need to simplify each equation and check if the resulting statement is true or false.
a. 4(x + 3) + 2x = 6(x + 2)
Simplifying both sides:
4x + 12 + 2x = 6x + 12
Combining like terms: 6x + 12 = 6x + 12
This is always true. It has infinite solutions.
b. 5 + 2(3 + 2x) = x + 3(x + 1)
Simplifying both sides:
5 + 6 + 4x = x + 3x + 3
Combining like terms: 4x + 11 = 4x + 3
This is always false. It has no solution.
c. 5(x + 3) + x = 4(x + 3) + 3
Simplifying both sides:
5x + 15 + x = 4x + 12 + 3
Combining like terms: 6x + 15 = 4x + 15
This is always true. It has infinite solutions.
d. 4 + 6(2 + x) = 2(3x + 8)
Simplifying both sides:
4 + 12 + 6x = 6x + 16
Combining like terms: 6x + 16 = 6x + 16
This is always true. It has infinite solutions.
Therefore, option (b) 5 + 2(3 + 2x) = x + 3(x + 1) has no solution.