Rational Expressions • Topic 6 of 6

Rational Equations

To solve an equation containing rational expressions, clear the fractions: multiply every term by the common denominator (or cross-multiply a single fraction equal to another), then solve the resulting equation. For 3/x = 1/2, cross-multiply to 6 = x. For 1/x + 1/2 = 1, subtract 1/2 to get 1/x = 1/2, so x = 2. Always exclude any value that makes a denominator zero, and check your solution in the original equation to reject extraneous roots. Rational equations combine fraction skills with equation-solving, a regular SAT Advanced Math task.

✅ Solved examples

1. Solve 3/x = 1/2.
Cross-multiply: 3·2 = x, so x = 6.
2. Solve 12/x = 4.
Multiply by x: 12 = 4x, so x = 3.
3. Solve 1/x + 1/2 = 1.
1/x = 1/2, so x = 2.
4. Solve x/(x − 2) = 3.
Multiply by (x − 2): x = 3x − 6, so 6 = 2x, x = 3.

✏️ Practice — try these, take hints as needed

1. Solve 4/x = 1/3.
Cross-multiply.
4·3 = x.
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12.
2. Solve 10/x = 5.
Multiply by x.
10 = 5x.
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2.
3. Solve 2/(x + 1) = 1.
Multiply by (x + 1).
2 = x + 1.
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1.
4. Solve 1/x + 1/3 = 2/3.
1/x = 2/3 − 1/3.
1/x = 1/3.
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3.
5. Solve 5/(2x) = 1/4.
Cross-multiply: 5·4 = 2x.
20 = 2x.
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10.

📝 Topic test — 8 questions

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