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The equation given is:

\[
|4x - 9| = 5x
\]

To solve this, we need to split it into two cases because of the absolute value.

### Case 1: \( 4x - 9 = 5x \)
Solve for \(x\):

\[
4x - 9 = 5x
\]

Subtract \(4x\) from both sides:

\[
-9 = x
\]

So, \(x = -9\).

### Case 2: \( 4x - 9 = -(5x) \)
Now, solve the negative case:

\[
4x - 9 = -5x
\]

Add \(5x\) to both sides:

\[
9x - 9 = 0
\]

Add 9 to both sides:

\[
9x = 9
\]

Divide both sides by 9:

\[
x = 1
\]

### Checking for extraneous solutions
We need to check both solutions by substituting them into the original equation to see if they work:

1. For \(x = -9\):

\[
|4(-9) - 9| = | -36 - 9| = |-45| = 45
\]
\[
5(-9) = -45
\]

Since \(45 \neq -45\), \(x = -9\) is not a valid solution.

2. For \(x = 1\):

\[
|4(1) - 9| = |4 - 9| = |-5| = 5
\]
\[
5(1) = 5
\]

Since both sides are equal, \(x = 1\) is a valid solution.

### Final Answer:
The only solution is \(x = 1\).

---

Would you like further details or have any questions?

Here are 5 related questions for further practice:
1. Solve the equation \( |2x + 3| = 7 \).
2. Solve \( |x^2 - 4| = x + 2 \) for all values of \(x\).
3. How do you solve the inequality \( |3x - 2| \leq 7 \)?
4. What happens if the absolute value equation has no solution? Provide an example.
5. Solve \( |5x - 1| = 3x + 7 \) and check for extraneous solutions.

**Tip**: Always check for extraneous solutions when solving absolute value equations since the absolute value function can introduce false solutions.

Solve the equation for all values of x: |4x - 9| = 5x.

Step-by-step solution to solve the absolute value equation |4x - 9| = 5x. This example covers the absolute value property and solving linear equations, suitable for high school students in Grades 9-12. Includes checks for extraneous solutions.

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