Question

question
solve the equation for all values of x.
|2x - 3| - 3 = 4x
answer
attempt 1 out of 2

Explanation:

Step1: Isolate the absolute value

Rearrange to isolate $|2x-3|$.
$|2x - 3| = 4x + 3$

Step2: Set up two cases

Case 1: Inside absolute value is non-negative.
$2x - 3 = 4x + 3$
Case 2: Inside absolute value is negative.
$2x - 3 = -(4x + 3)$

Step3: Solve Case 1

Rearrange and solve for $x$.
$-2x = 6 \implies x = -3$

Step4: Check Case 1 validity

Verify if $4x+3\geq0$ (RHS of isolated absolute value).
$4(-3)+3=-9<0$, so $x=-3$ is invalid.

Step5: Solve Case 2

Expand and solve for $x$.
$2x - 3 = -4x - 3 \implies 6x=0 \implies x=0$

Step6: Check Case 2 validity

Verify if $4x+3\geq0$.
$4(0)+3=3\geq0$, so $x=0$ is valid.

Answer:

$x=0$