SAT Rational Equations: Fractions, Restrictions & Practice

A rational expression contains a variable in a denominator.

Example:

[ \frac{3}{x-2} ]

The most important rule:

A denominator can never equal zero.

So:

[ x\ne2 ]

Simple proportion

[ \frac{x}{5}=\frac{12}{15} ]

Cross-multiply:

[ 15x=60 ]

[ x=4 ]

Clearing denominators

Solve:

[ \frac{x}{3}+\frac{x}{6}=6 ]

Multiply everything by 6:

[ 2x+x=36 ]

[ 3x=36 ]

[ x=12 ]

Once denominators are cleared, you're often left solving a straightforward linear equation — see the rest of our SAT Math guides for other core skills.

Variable in the denominator

Solve:

[ \frac{4}{x}=2 ]

Restriction:

[ x\ne0 ]

Multiply by x:

[ 4=2x ]

[ x=2 ]

Valid.

Multiple rational terms

[ \frac{1}{x}+\frac{1}{2}=\frac{3}{4} ]

Multiply by (4x):

[ 4+2x=3x ]

[ x=4 ]

Rational expression simplification

[ \frac{x^2-9}{x-3} ]

Factor:

[ \frac{(x-3)(x+3)}{x-3}=x+3 ]

but the original expression still excludes:

[ x=3 ]

That restriction does not disappear.

That factoring instinct is the same one you'll use throughout quadratic equations.

Word problem example

A machine completes (1/6) of a job per hour.

Another completes (1/4) per hour.

Together:

[ \frac16+\frac14=\frac{2}{12}+\frac{3}{12}=\frac{5}{12} ]

job per hour.

Time for one job:

[ \frac{1}{5/12}=\frac{12}{5}=2.4 ]

hours.

Work-rate problems like this one sometimes expand into a full system of equations when there's more than one unknown rate.

Original practice

1

Solve:

[ \frac{x}{8}=\frac{9}{12} ]

2

Solve:

[ \frac{2}{x}=5 ]

3

Simplify:

[ \frac{x^2-16}{x-4} ]

4

What value is excluded from:

[ \frac{7}{x+5} ]

5

Solve:

[ \frac{x}{4}+\frac{x}{2}=9 ]

Answers

1. 6
2. 2/5
3. (x+4), with (x\ne4)
4. -5
5. 12

Common traps

• cross-multiplying when expressions are not set as one fraction equal to one fraction;
• forgetting denominator restrictions;
• canceling terms across addition;
• keeping an extraneous solution.

Bottom line

For rational equations:

1. note restrictions;
2. find the least common denominator;
3. clear fractions;
4. solve;
5. check.

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