How to solve a rational equation with like denominators

Key concept: When both fractions share the same denominator, it can feel like the problem should collapse fast. It does — but only if you keep the denominator restriction in mind while simplifying.

Step by step

Start by writing the equation clearly

The equation is intended to compare rational expressions with denominator $x-9$.

A useful move is to combine everything on one side, then clear the denominator.

Step 1: Bring terms together

After moving terms, the work should lead to a single fraction over $x-9$.

Step 2: Combine the numerators

When the fractions have the same denominator, you can combine them directly:

$\frac{-3x}{x-9}-\frac{5x+72}{x-9}=\frac{-3x-(5x+72)}{x-9}=\frac{-8x-72}{x-9}$

Step 3: Compare with the right side

If this equals $-4$, then multiply both sides by $(x-9)$:

$-8x-72=-4(x-9)$

Expand the right side:

$-8x-72=-4x+36$

Step 4: Solve

Add $8x$ to both sides:

$-72=4x+36$

Subtract 36:

$-108=4x$

So

$x=-27$

Step 5: Check the restriction

Since $x-9\ne 0$, the only forbidden value is

$x\ne 9$

So $x=-27$ is valid.

Final answer

$\boxed{x=-27}$

Pitfall alert

A lot of errors come from combining fractions too aggressively. Keep the numerator signs straight: $-(5x+72)$ is not the same as $-5x+72$. That sign slip changes the whole answer.

Try different conditions

If the right side were a different constant, the same strategy still works: combine the fractions, clear the denominator, and solve the resulting linear equation. If the algebra produced $x=9$, that would still be invalid because the denominator would vanish.

Further reading

common denominator, linear rational equation, domain check