Rational Expressions • Topic 5 of 6

Subtraction

Subtracting rational expressions follows the same common-denominator rule, with extra care for the minus sign, which applies to the entire second numerator. With equal denominators, subtract the tops: 5/x − 2/x = 3/x. With different denominators, rewrite over the least common denominator first. When the second numerator has more than one term, put it in brackets so the subtraction distributes correctly. After subtracting, combine like terms and simplify. Sign errors when distributing the minus are the most common SAT slip here, so brackets and a final check are worthwhile.

✅ Solved examples

1. Simplify 5/x − 2/x.
Same denominator: (5 − 2)/x = 3/x.
2. Simplify 1/x − 1/(2x).
Common denominator 2x: 2/(2x) − 1/(2x) = 1/(2x).
3. Simplify 5/(x + 1) − 2/(x + 1).
Same denominator: 3/(x + 1).
4. Simplify 1/x − 1/(x + 1).
Common denominator x(x + 1): [(x + 1) − x]/[x(x + 1)] = 1/[x(x + 1)].

✏️ Practice — try these, take hints as needed

1. Simplify 9/x − 4/x.
Same denominator.
Subtract numerators.
—
5/x.
2. Simplify 7/(2x) − 3/(2x).
Same denominator.
(7 − 3)/(2x) = 4/(2x).
Reduce.
2/x.
3. Simplify 1/x − 1/(3x).
Common denominator 3x.
3/(3x) − 1/(3x).
—
2/(3x).
4. Simplify 8/(x − 1) − 3/(x − 1).
Same denominator.
Subtract numerators.
—
5/(x − 1).
5. Simplify 10/x − 6/x.
Same denominator.
(10 − 6)/x.
—
4/x.

📝 Topic test — 8 questions

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