Question

Solve for the unknown:
|2x-5|x-2|| = |x+6| + 3

Answer 1

what a pain

Answer 2

let me see if i can work this out on pencil and paper before writing the solution.

Answer 3

sure. We know it's a pain, even with 4 brains already working on it. Thanks for the help.

Answer 4

i think you have to work in cases. for example the right hand side is a piecewise function, namely
\[ |x+6|+3 = \left\{
\begin{array}{lr}
x+9 & : x >6\\
-x-3 & : x < 6
\end{array}
\right.\]

Answer 5

left hand side is more of a pain

Answer 6

ok i have one of the solutions.

Answer 7

shoot. :)

Answer 8

the left hand side is |2x-5|x-2||

Answer 9

now if x > 2 this is
|2x-5(x-2)| = |2x-5x+10| = |-3x+10| yes?

Answer 10

yes.

Answer 11

because of course if x > 2 then |x-2|=x-2

Answer 12

and then?

Answer 13

now |-3x+10| = -3x + 10 if x < 10/3 and
|-3x+10|=10-3x if x > 10/3

Answer 14

so if x> 6 both conditions hold and we have the equation
\[10-3x=x+9\] solve for x to get
\[x=\frac{19}{2}\]

Answer 15

that is if x > 6 then the left hand side is 10-3x and the right hand side is x+9 so solve for x and that is one answer. now we have to do it again for the other side

Answer 16

go on. :)

Answer 17

again on the left if x < 2 then |2x-5|x-2|| is
|2x-5(2-x)| is
|2x-10+5x| is
|7x-10|

Answer 18

hope these steps are clear

Answer 19

i am just replacing |x-2| with (2-x) for x < 2

Answer 20

yes yes. go on

Answer 21

and |7x-10| = 7x-10 if x > 10/7
and |7x-10|=10-7x if x < 10/7

Answer 22

now for x < 10/7 still x > -6 so set

\[10-7x=x+9\]
\[1=8x\]
\[x=\frac{1}{8}\]

Answer 23

and those are the two solutions

Answer 24

what a pain!

Answer 25

btw i made a typo on the first line

Answer 26

yes we got that

Answer 27

*bows* :))) thanks man, that was your second medal :D

Answer 28

i was so impressed with myself for writing a piecewise function that i wrote it incorrectly. it should be
\[|x+6|+3 = \left\{
\begin{array}{lr}
x+9 & : x >-6\\
-x-3 & : x < -6
\end{array}
\right.\]

Answer 29

i had 6 instead of -6

Answer 30

yw

Answer 31

i guess while i am showing off that you should say the line
\[7x-10\] for x between 10/7 and 2 does not intersect the line x+9

Answer 32

anyway, we got the main gist of your solution, and was able to work it out. thanks again :-bd