Question

How do you solve `abs(5x+3)=abs(2x-1)`?

Answer 1

Case 1: Both Absolute Values are positive

`5x + 3 = 2x - 1`

`3x = -4`

`x = -4/3`

Case 2: The left absolute value is negative

`-(5x + 3) = 2x - 1`

`-5x - 3 = 2x - 1`

`-7x = 2`

`x = -2/7`

Case 3: The right absolute value is negative

`5x + 3 = -(2x - 1)`

`5x + 3 = -2x + 1`

`7x = -2`

`x = -2/7`

As you can see, whether the absolute value on right is negative and the other positive or the absolute value on left is negative and the other positive, both scenarios will give the same result.

Let's just check our solutions to make sure none are extraneous.

`|5(-4/3) + 3| =^? |2(-4/3) - 1|`

Doing all the fractal operations on this, you will find that this, as well as the other solution, `x = -2/7`, work in the original equation.

Hence, our solution set is `{x = -2/7, -4/3}`.

Hopefully this helps!