Equations with algebraic fractions · Grade 9
In Equations with algebraic fractions (Grade 9) you practise 4 skills: Equations with numbers in the denominator · Values x cannot take · Solving equations with algebraic fractions · A solution that has to be rejected. In an equation with an algebraic fraction, the unknown appears in a denominator. You cannot divide by zero, so first find the values x cannot take: those that make a denominator 0. Factorise a quadratic denominator first.
To solve, multiply both sides by the common denominator, so that the fractions disappear. Then solve the equation you get in the usual way. Last, check every solution against the forbidden values. A solution that makes a denominator 0 is rejected. If every solution is rejected, the equation has no solution.
Tip: Write down the forbidden values before you multiply, and check against them at the end.
What you learn
Equations with numbers in the denominator
For example: Solve: x ÷ 10 + 7 = 4
Answer and solution
The answer: −30
Subtract 7 from both sides: x ÷ 10 = −3 Multiply both sides by 10: x = −3 × 10 = −30
Values x cannot take
For example: Which value of x is not allowed in 6 ÷ (2x + 14) ?
Answer and solution
The answer: −7
2x + 14 = 0 , so x = −7
Solving equations with algebraic fractions
For example: Solve: 36 ÷ (x − 3) = −6
Answer and solution
The answer: −3
Not allowed: x = 3 . Multiply both sides by the denominator x − 3 : 36 = −6 × (x − 3) 36 = −6x + 18 −6x = 18 x = −3 The solution is allowed: it makes no denominator 0.
A solution that has to be rejected
For example: What is the solution of (x − 7) ÷ (x − 9) = 2 ÷ (x − 9) ?
- No solution
- x = 9
Answer and solution
The answer: No solution
Not allowed: x = 9 , because then the denominator is 0. Multiply both sides by the denominator: x − 7 = 2 x = 9 But that is exactly the forbidden value, so the equation has no solution.
Not on the site yet
Parts of this topic that are not built yet: Equations with algebraic fractions and a parameter · Inequalities with algebraic fractions.
The exercises are made by rules and the answers are worked out in code, so there are new exercises every time. The topic follows the Israeli Ministry of Education's grade 9 maths curriculum (2026–27).