Rational Expressions • Topic 4 of 6

Addition

To add rational expressions, write them over a common denominator, add the numerators, then simplify. With the same denominator, just add the tops: 2/x + 3/x = 5/x. With different denominators, the least common denominator is their product (or LCM): 1/x + 1/(2x) uses 2x, giving 2/(2x) + 1/(2x) = 3/(2x). To add a whole number, write it over the denominator: 1/x + 1 = (1 + x)/x. Combine like terms in the numerator at the end. Common-denominator setup is the whole skill, and the SAT tests it directly and inside equations.

✅ Solved examples

1. Simplify 2/x + 3/x.
Same denominator: (2 + 3)/x = 5/x.
2. Simplify 1/x + 1/(2x).
Common denominator 2x: 2/(2x) + 1/(2x) = 3/(2x).
3. Simplify 1/x + 1.
Write 1 as x/x: (1 + x)/x.
4. Simplify 1/(x − 1) + 1/(x + 1).
Common denominator (x² − 1): [(x + 1) + (x − 1)]/(x² − 1) = 2x/(x² − 1).

✏️ Practice — try these, take hints as needed

1. Simplify 4/x + 5/x.
Same denominator.
Add numerators.
—
9/x.
2. Simplify 1/x + 3/(2x).
Common denominator 2x.
2/(2x) + 3/(2x).
—
5/(2x).
3. Simplify 2/x + 3.
Write 3 as 3x/x.
(2 + 3x)/x.
—
(2 + 3x)/x.
4. Simplify 3/(x + 2) + 2/(x + 2).
Same denominator.
Add numerators.
—
5/(x + 2).
5. Simplify x/4 + x/4.
Same denominator.
(x + x)/4.
—
x/2.

📝 Topic test — 8 questions

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