Question

Solve absolute value equation. Check for extraneous solution. |2x-3|=2x-1

Answer 1

`2x-3 < 0\ \ \ |+3\\2x < 3\ \ \ |:2\\x < 1.5\\\\for\ x\in(-\infty;\ 1.5),\ 2x-3 < 0\to|2x-3|=-(2x-3)=-2x+3\\\\for\ x\in\left[1.5;\ \infty\right),\ 2x-3\geq0\to|2x-3|=2x-3\\\\(1)\ x\in(-\infty;\ 1.5)\\\\-2x+3=2x-1\ \ \ \ |-3\\-2x=2x-4\ \ \ |-2x\\-4x=-4\ \ \ \ |:(-4)\\x=1\in(-\infty;\ 1.5)\\\\(2)\ x\in[1.5;\ \infty)\\\\2x+3=2x-1\ \ \ \ |-2x\\3=-1\ FALSE\to x\in\O\\\\Answer:\ x=1`

Answer 2

|2x-1|=2x-1 translates to two equations:

2x-3=-(2x-1) and 2x-3=2x-1

2x-3=-(2x-1)

2x-3=-2x+1

4x=4

x=1

now second equation

2x-3=2x-1

0=4, no solution.

So only value for x is 1. Now plug in x to check if it works or not.

|2(1)-1|=2(1)-1

|2-1|=2-1

2-1=1

1=1

so x can only equal 1.

x=1