Question

|x-3|+|2x-8|=5. Help me to solve this problem about equation please?

Answer 1

`x = {2,16/3}`

Explanation:

This equation can also be stated as

`sqrt((x-3)^2)+sqrt((2x-8)^2)=5` and squaring both sides

`(x-3)^2+(2x-8)^2+2sqrt((x-3)^2)sqrt((2x-8)^2)=25`

Arranging and squaring again

`4(x-3)^2(2x-8)^2 = (25-((x-3)^2+(2x-8)^2))^2` or

`4(x-3)^2(2x-8)^2-(25-((x-3)^2+(2x-8)^2))^2=0` or

`3 (x-10) (x-2) x (3 x-16) = 0` and the potential solutions are

`x = {0,2,10,16/3}` and the feasible solutions are

`x = {2,16/3}` because they verify the original equation.

Answer 2

`x=16/3 or x =2`

Explanation:

`|x-3|+|2x-8|=5`
Start by adding `color(red)(-|2x-8|` to both sides.
`|x-3|cancel(+|2x-8|) cancelcolor(red)(-|2x-8|)= 5 color(red)(-|2x-8)`
`|x-3| = - |2x -8|+5`
We know....
Either `x - 3= -|2x -8|+5` or `x -3= -(-|2x-8|+5)`
Let's start with part `1`
`x - 3 = - |2x-8|+5`
Flip the equation to fill more comfortable
`-|2x -8|+5= x-3`
We want to eliminate `5` on the left side and transfer it to the other side, to do that, we need to add `color(red)(-5)` to both sides
`-|2x-8|cancel(+5) cancelcolor(red)(-5) = x -3 color(red)(-5)`
`-|2x -8|=x-8`
We need to cancel the negative sign in front of the absolute value. To do that, we need to divide both sides by `color(red)(-1)`
`(-|2x-8|)/color(red)(-1) = (x-8)/color(red)(-1)`
`|2x-8|=-x+8`
We know either `2x -8=x-8 or 2x -8= -(-x+8)`
Let's start with the first possibility.
`2x - 8 = -x +8`
Start by adding `color(red)(x)` to both sides
`2x -8+color(red)x =x+8+color(red)(x)`
`3x - 8 = 8`
`3x = 8 + 8`
`3x = 16`
`x = 16/3`
Solve for the second possibility
`2x - 8 = - (-x + 8)`
`2x - 8 = x - 8`
Combine like terms
`2x - x = -8 + 8`
`x = 0` (doesn't work in original equation)

Part 2:
`x - 3 = -(-|2x -8|+5)` (Look at the first one to see what I'mtalking about)
Flip the equation
`|2x -8|-5=x-3` (transfer 5 on the right side)
`|12x - 8|= x -3 + 5`
`|12x-8| = x + 2`
We know either `2x - 8=x+2 or 2x-8= -(x+2)`
Let's start solving the first possibility
`2x -8=x+2`
Combine like terms
`2x - x = 2 + 8`
`x = 10`
Solve the second possibility
`2x -8=-(x+2)`
`2x - 8= -x - 2`
Combine like terms
`2x + x = -2 + 8`
`3x = 6`
`x = 6/3`
`=2` (Works in original equation)

Thus,
The final answer are `x=16/3 or x=2`