Answer
$x = -1$
Work Step by Step
We use the definition of absolute value to rewrite the equation as two separate equations:
$5x - 2 = 7x + 14\quad$ or $\quad5x - 2 = - (7x + 14)$
Use distributive property to simplify the second equation:
$5x - 2 = 7x + 14\quad$ or $\quad5x - 2 = -7x - 14$
Add $2$ to both sides of the equation:
$5x = 7x + 16\quad$ or $\quad5x = -7x - 12$
Subtract $7x$ from each side of the equation:
$-2x = 16\quad$ or $\quad12x = -12$
Divide the first equation by $-2$ and the second equation by $12$ to isolate $x$:
$x = -8\quad$ or $\quad x = -1$
Now we can check for extraneous solutions by plugging each solution back into the original equation.
Let's check $x = -8$ first:
$|5(-8) - 2| = 7(-8) + 14$
Multiply first:
$|-40 - 2| = -56 + 14$
Add or subtract:
$|-42| = -42$
This solution is extraneous because the absolute value of a number cannot be negative.
Let's check $x = -1$:
$|5(-1) - 2| = 7(-1) + 14$
Multiply first:
$|-5 - 2| = -7 + 14$
Add or subtract:
$|-7| = 7$
This is the only correct solution.
