To solve the equation $|5x - 9| = 4x$, we need to consider the two possible cases of the absolute value expression.

Case 1: When $5x - 9 \geq 0$

In this case, the absolute value can be removed as: $5x - 9 = 4x$ Solving this: $5x - 4x = 9$ $x = 9$

Now we check whether this solution satisfies the condition $5x - 9 \geq 0$: $5(9) - 9 = 45 - 9 = 36 \geq 0$ So, $x = 9$ is a valid solution for this case.

Case 2: When $5x - 9 < 0$

In this case, the absolute value changes sign: $-(5x - 9) = 4x$ Simplifying: $-5x + 9 = 4x$ $9 = 9x$ $x = 1$

Now we check whether this solution satisfies the condition $5x - 9 < 0$: $5(1) - 9 = 5 - 9 = -4 < 0$ So, $x = 1$ is also a valid solution for this case.

Final Answer:

The solutions to the equation are $x = 1$ and $x = 9$.

Do you have any questions or would you like more details?

Here are some related questions:

1. How would you solve this equation if the right-hand side were different?
2. What happens if the absolute value expression is more complicated?
3. Can you graph these solutions and the original equation?
4. How would you solve this equation if the absolute value was on both sides?
5. What if $4x$ is replaced by a constant?

Tip: When solving absolute value equations, always consider both the positive and negative cases of the expression.