Raj wrote the four equations that are shown below.

Question

A

3 (x + 4) = 3 (x + 2) + 6
B

6 (x minus 1) = 3 (x minus 1) + 3

C

5 x + 4 minus x = 2 (2 x minus 2)
D

Negative 2 (x + 3) + x = negative x minus 6

He examined them, without solving them, to determine which has no solution. Which of Raj’s equations has no solution?
3 (x + 4) = 3 (x + 2) + 6
6 (x minus 1) = 3 (x minus 1) + 3
5 x + 4 minus x = 2 (2 x minus 2)
Negative 2 (x + 3) + x = negative x minus 6

Answer 1

To determine which of Raj's equations has no solution, we can analyze each equation and see if it leads to a contradiction or an impossible scenario.

Equation A:

\[ 3(x + 4) = 3(x + 2) + 6 \]

Expanding both sides: \[ 3x + 12 = 3x + 6 + 6 \] \[ 3x + 12 = 3x + 12 \]

This simplifies to a true statement, which means there are infinitely many solutions (any x works).

Equation B:

\[ 6(x - 1) = 3(x - 1) + 3 \]

Expanding both sides: \[ 6x - 6 = 3x - 3 + 3 \] \[ 6x - 6 = 3x \]

Subtracting \(3x\) from both sides: \[ 3x - 6 = 0 \] \[ 3x = 6 \] \[ x = 2 \]

This equation has a unique solution \(x = 2\).

Equation C:

\[ 5x + 4 - x = 2(2x - 2) \]

Simplifying the left side: \[ 4x + 4 = 4x - 4 \]

Subtracting \(4x\) from both sides: \[ 4 = -4 \]

This statement is a contradiction, which means there are no solutions for this equation.

Equation D:

\[ -2(x + 3) + x = -x - 6 \]

Expanding the left side: \[ -2x - 6 + x = -x - 6 \] \[ -x - 6 = -x - 6 \]

This simplifies to a true statement as well, which means there are infinitely many solutions (any x works).

Conclusion:

The equation that has no solution is Equation C: \[ 5x + 4 - x = 2(2x - 2) \]