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

2 Answers
Oct 23, 2017

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.

Oct 23, 2017

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