Answer
$\text{a. The equation is never true.}$
$\text{b. The equation is always true.}$
Work Step by Step
a. Group like terms on both sides of the equation:
$(7x - 4x) + 6 = (12 - 8) + 3x$
Combine like terms:
$3x + 6 = 4 + 3x$
Subtract $6$ from each side of the equation to move constants to the right side of the equation:
$3x = -2 + 3x$
Subtract $3x$ from each side of the equation:
$0 = -2$
The two sides are not equal to one another; therefore, no value of $x$ will ever make the original equation true. So, the equation has no solution and is never true.
b. Use the distributive property for subtraction to simplify both sides of the equation:
$2x + (3x - 12) = (4x - 12) + x$
Group like terms:
$(2x + 3x) - 12 = (4x + x) - 12$
Combine like terms:
$5x - 12 = 5x - 12$
Add $12$ to each side of the equation to isolate constants on one side of the equation:
$5x = 5x$
Subtract $5x$ from each side of the equation:
$0 = 0$
Both sides of the equation are equal; therefore, any value of $x$ will make the original equation true. So, the equation is always true.
