The question is simple:

Find the value(s) for x from the following equation.

abs(x) = 4x + 3

When we solve the problem by taking the +ve and -ve values for x. We get -3/5 and -1.

But substituting -3/5 satisfies the equations, substituting -1 does not.

What am I missing here, please help.

Its because -1 doesn’t satisfy your initial condition of |x| = 4x+3, Many people miss this step of checking if their solution satisfy the original condition asked in the question.

OR

Let me explain it better :

When you open modulo , you open it on the following assumptions:

For Positive :

x = 4x + 3 (for \ x \geq 0)

-3 = 3x \ or \ x = -1

But your initial assumption was x greater than equal to zero hence, we reject this solution

For Negative :

-x = 4x + 3 (for \ x \lt 0)

-3 = 5x

x = -\frac{3}{5}, \text{Which is less than 0, hence it satisfy } \ x \lt 0

For further clarification checkout this : https://gmatclub.com/forum/math-absolute-value-modulus-86462.html