Fixing a rational-equation simplification error

Key takeaway: This is the kind of work where one sign error can send everything sideways. The cleanest way to debug it is to go back to the original equation and combine terms with care.

Key idea

The expression

$4-\frac{3x}{x-9}=\frac{5x-72}{x-9}$

has the same denominator on both fractions, so the safest move is to multiply through by $(x-9)$.

Step 1: Multiply both sides by $(x-9)$

$4(x-9)-3x=5x-72$

Step 2: Expand

$4x-36-3x=5x-72$

Step 3: Combine like terms

$x-36=5x-72$

Step 4: Solve

Add 72 to both sides:

$36=4x$

So

$x=9$

Step 5: Check the domain

But the original equation is undefined at $x=9$ because the denominator becomes zero:

$x-9=0$

So $x=9$ must be rejected.

Final answer

$\boxed{\text{No solution}}$

---

Pitfalls the pros know 👇 Don’t trust a candidate answer just because the arithmetic works. In rational equations, the last line of defense is always substitution into the original expression, not the simplified one.

What if the problem changes? If the denominator were $x-8$ instead of $x-9$, the same algebra would likely produce a different candidate and a different validity check. The method stays the same; only the forbidden values change.

Tags: domain restriction, extraneous root, rational equation