Question

How do you solve `abs(4x+3)=2x-1`?

Answer 1

Solve |4x + 3| = 2x - 1

Explanation:

Answer deleted

Answer 2

Split into cases `4x + 3 >= 0` and `4x + 3 < 0` then solve the resulting linear equations.

We find that the resulting solutions are both spurious, so `abs(4x+3) = 2x-1` has no solutions.

Explanation:

Split into cases `4x+3 >= 0` and `4x+3 < 0`

Case 1 : 4x+3 >= 0

If `4x+3 >= 0` then `abs(4x+3) = 4x+3`

Solve `4x+3 = 2x-1`

Subtract `2x` from both sides to get:

`2x+3 = -1`

Subtract #3 from both sides to get:

`2x = -4`

Divide both sides by `2` to get `x = -2`

Then `4x + 3 = 4(-2)+3 = -8 + 3 = -5 < 0`

So `x = -2` is a spurious solution, since it does not satisfy `4x+3 >= 0`

Case 2 : 4x+3 < 0

If `4x+3 < 0` then `abs(4x+3) = -(4x+3)`

Solve `-(4x+3) = 2x - 1`

Add `4x` to both sides to get:

`-3 = 6x - 1`

Add `1` to both sides to get:

`-2 = 6x`

Divide both sides by `6` to get `x = -1/3`

Then `4x+3 = -4/3+3 = -4/3+9/3 = 5/3 > 0`

So `x = -1/3` is a spurious solution, since it does not satisfy `4x+3 < 0`