Could someone please tell me how to solve the following without graphing?
(1) $\displaystyle |x+4|+|x-7|=|2x-1|$
(2) $\displaystyle |x-|x-1||=\lfloor x \rfloor$ (Greatest integer function of x)
Thanks in advance!
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.
I hope this was helpful
For $\displaystyle x \geq 7$:
$\displaystyle |x + 4| + |x - 7| = |2x - 1|$ is equivalent to
$\displaystyle x + 4 + x - 7 = 2x - 1$
$\displaystyle 2x - 3 = 2x - 1$
which is impossible.
For $\displaystyle 0.5 \leq x < 7$:
$\displaystyle |x + 4| + |x - 7| = |2x - 1|$ is equivalent to
$\displaystyle x + 4 + 7 - x = 2x - 1$
$\displaystyle 11 = 2x - 1$
$\displaystyle 12 = 2x$
$\displaystyle x = 6$
and that's one solution.
For $\displaystyle -4 \leq x < 0.5$:
$\displaystyle |x + 4| + |x - 7| = |2x - 1|$ is equivalent to
$\displaystyle x + 4 + 7 - x = 1 - 2x$
$\displaystyle 11 = 1 - 2x$
$\displaystyle 10 = -2x$
$\displaystyle x = -5$
but this solution is not in the correct domain.
For $\displaystyle x < -4$:
$\displaystyle |x + 4| + |x - 7| = |2x - 1|$ is equivalent to
$\displaystyle -x - 4 + 7 - x = 1 - 2x$
$\displaystyle -2x + 3 = -2x + 1$
which is impossible.
Hence, the one solution to this equation is x = 6.