Algebraic fractions

A fraction with x living in it

You know fractions. You know expressions with a letter. An algebraic fraction is both at once: a fraction with an unknown in it — often right in the denominator.

Like 6/x. Or 5/(x − 3). It looks scary, but the rules are the same as for ordinary fractions. With one new bit.

That new bit is called a condition. You must not divide by zero — that has been true since year 1. So the denominator must never come out as zero. For 6/x, x must not be 0. For 5/(x − 3), x must not be 3, because 3 − 3 = 0.

You write the condition first, before you start calculating. It is like a no-entry sign: one particular number just must not go in.

In year 9 and in entrance exams, conditions get marks. Who forgets them loses points even for an otherwise correct calculation. Today you will learn to find conditions and cancel algebraic fractions.

Tereza is doing homework and calls Jonáš: “I got x + 2 and it is supposedly wrong. But I cancelled correctly!” Jonáš looks at the question: (x² + 2x)/x. “The cancelling is fine. But you are missing the condition x ≠ 0. If you put zero in for x, you divide by zero — and that does not work.” Tereza groans: “So the whole calculation is right and a mark down because of one line?” Exactly. The condition is the line you write first and forget most often.

Conditions and cancelling step by step

Step 1: Find the denominator. That is the bottom of the fraction. For 8/(x − 5) the denominator is x − 5.

Step 2: Set the denominator equal to zero and solve. x − 5 = 0, so x = 5. That number is forbidden.

Step 3: Write the condition. You write: x ≠ 5. Read ≠ as “must not equal”.

Step 4: Only now go on calculating — for example cancel. Cancelling means divide the numerator and the denominator by the same expression. With ordinary fractions you cancel with numbers: 6/8 = 3/4 (divided by two). With algebraic fractions you cancel letters too.

Example: (3x)/(6x). There is 3x on top and on the bottom. Cancel: you get 1/2. Condition: x ≠ 0.

Watch one rule: you may only cancel factors, not terms you add. A factor is what you multiply. For (2x + 4)/2 you cannot just cross out the four — first factor out: 2x + 4 = 2 · (x + 2). Then (2 · (x + 2))/2 = x + 2. The two cancelled fully, because it multiplied the whole bracket.

Example 1: the condition for 7/(x − 3)

Question: Find the condition for when the expression 7/(x − 3) makes sense.

Method: The denominator is x − 3. Set it equal to zero: x − 3 = 0. Solve: x = 3. That is the forbidden number — if x were 3, you would divide by zero.

Write: x ≠ 3.

Sense check: Put in x = 4: the denominator is 1, the expression is 7. Fine. Put in x = 3: the denominator is 0 and 7/0 does not exist. The condition holds.

Example 2: cancel (2x + 6)/2

Question: Simplify the expression (2x + 6)/2.

Method: On top there is a sum, so you must not cancel just a piece. First factor out the two: 2x + 6 = 2 · (x + 3).

The expression now looks like this: (2 · (x + 3))/2. The two is a factor on top and on the bottom, so you can cancel it.

Result: x + 3.

Check by substituting: For x = 1 the original expression gives (2 + 6)/2 = 4. The new expression gives 1 + 3 = 4. It matches. The denominator was only two, no x — you do not need a condition here.

Example 3: cancel (x² + 5x)/x

Question: Simplify (x² + 5x)/x and find the condition.

Method: Condition first: the denominator is x, so x ≠ 0.

On top, factor out x: x² + 5x = x · (x + 5).

The expression is (x · (x + 5))/x. Cancel x on top and on the bottom.

Result: x + 5, condition x ≠ 0.

Check: For x = 2 the original expression gives (4 + 10)/2 = 7. The new one gives 2 + 5 = 7. It matches. And zero stays forbidden, even if there is no fraction left in the result — the condition belongs to the original question.

Practice: now you do it 🎲

Now try it for real. You will see all the questions at once, like on paper. Bắt đầu with ten. When it goes well, add more.

Practise substituting into expressions — with algebraic fractions you will use it as a check. Write whole numbers.

Practise the questions

On paper

Conditions and cancelling. For every expression write the condition as the first line.

1. Find the conditions for the expressions: 5/x, 3/(x − 7), 10/(x + 2). For each write the forbidden number with the sign ≠.

2. Cancel: (3x + 9)/3, (5x + 10)/5, (4x + 8)/4. Always factor out first, then cross out.

3. Cancel (x² + 7x)/x. Do not forget the condition.

4. Check one result by substituting: pick x = 2, count the original expression and the cancelled one. They must come out the same.

Now you 💪

1. You can say why the denominator must not be zero, and find the forbidden number for 4/(x − 6).
2. You cancelled (3x + 9)/3 to x + 3 — by factoring out, not by crossing out the three next to the nine.
3. You checked at least one result by putting a number in, and it matched.

What to take from this lesson

• An algebraic fraction is a fraction with an unknown in the denominator, like 6/x.
• The denominator must not be zero — write the condition (x ≠ number) as the first line.
• You find the forbidden number with an equation: denominator = 0.
• You may only cancel factors. If there is a sum on top, factor out first, then cancel.
• A check by substituting spots bad cancelling in ten seconds.

© 2026 Ing. Martin Polak / AlgoRhino · Điều khoản sử dụng nội dung