1. ## Absolute Value Equations

Could someone please tell me how to solve the following without graphing?

(1) $|x+4|+|x-7|=|2x-1|$

(2) $|x-|x-1||=\lfloor x \rfloor$ (Greatest integer function of x)

2. Originally Posted by Winding Function
Could someone please tell me how to solve the following without graphing?

(1) $|x+4|+|x-7|=|2x-1|$

(2) $|x-|x-1||=\lfloor x \rfloor$ (Greatest integer function of x)

For number (1) if you state the boundaries for what the absolute value signs would do to the terms you can solve it as a simple algebraic equation.

For example: |x-7| take away the absolute value signs and state that it is only true for positive real numbers x-7 > or equal to 0

|x+4| is just x+4 because the absolute value sign doesn't affect positive real numbers.

3. Um, the problem didn't specify any boundaries. Could someone please show me how to solve the equations? Thanks!

4. For $x \geq 7$:

$|x + 4| + |x - 7| = |2x - 1|$ is equivalent to

$x + 4 + x - 7 = 2x - 1$
$2x - 3 = 2x - 1$
which is impossible.

For $0.5 \leq x < 7$:

$|x + 4| + |x - 7| = |2x - 1|$ is equivalent to

$x + 4 + 7 - x = 2x - 1$
$11 = 2x - 1$
$12 = 2x$
$x = 6$
and that's one solution.

For $-4 \leq x < 0.5$:

$|x + 4| + |x - 7| = |2x - 1|$ is equivalent to

$x + 4 + 7 - x = 1 - 2x$
$11 = 1 - 2x$
$10 = -2x$
$x = -5$
but this solution is not in the correct domain.

For $x < -4$:

$|x + 4| + |x - 7| = |2x - 1|$ is equivalent to

$-x - 4 + 7 - x = 1 - 2x$
$-2x + 3 = -2x + 1$
which is impossible.

Hence, the one solution to this equation is x = 6.