|x|-4x=15

To solve the equation \(|x| - 4x = 15\), we need to consider two cases based on the definition of the absolute value.

Case 1: \(x \geq 0\)

In this case, \(|x| = x\). Therefore, the equation becomes:

\[ x - 4x = 15 \]

This simplifies to:

\[ -3x = 15 \]

Dividing both sides by -3 gives:

\[ x = -5 \]

Since \(-5\) is not greater than or equal to \(0\), this solution is not valid for this case.

Case 2: \(x < 0\)

In this case, \(|x| = -x\). Therefore, the equation becomes:

\[ -x - 4x = 15 \]

This simplifies to:

\[ -5x = 15 \]

Dividing both sides by -5 gives:

\[ x = -3 \]

This solution \(x = -3\) is valid since it satisfies the condition \(x < 0\).

Conclusion

The solution to the equation \(|x| - 4x = 15\) is:

\[ \boxed{-3} \]