Rational Expressions and Equations

Rational Expressions on the Exam

Rational expressions—polynomials divided by polynomials—appear throughout Praxis 5165 simplification and equation-solving items. They connect factoring skills to domain awareness, a recurring theme when ETS asks which value is excluded from the domain.

Domain Restrictions

For P(x)/Q(x), exclude values where Q(x) = 0. Simplified form may hide original restrictions; keep all excluded values from the original denominator.

Worked Example 1 — Domain

f(x) = (x + 1)/(x^2 − 9). Factor denominator: x ≠ 3, −3.

Simplifying

Factor, cancel common factors (not separate terms).

Worked Example 2

For x ≠ 0 and x ≠ 3, simplify (x^2 − 9)/(x^2 − 3x).

Numerator: (x − 3)(x + 3). Denominator: x(x − 3).

Cancel x − 3: (x + 3)/x, with x ≠ 0 and x ≠ 3 still stated.

Trap: Canceling terms in (x + 3)/x is invalid—only factors common to numerator and denominator cancel.

Multiplying and Dividing

Multiply across; divide by multiplying the reciprocal.

Worked Example 3

(x^2 − 1)/(x + 2) · (x + 2)/(x − 1) = (x − 1)(x + 1)/(x + 2) · (x + 2)/(x − 1) = x + 1 (x ≠ −2, 1).

Worked Example 4

(x^2 − 4x)/(x − 1) ÷ (x − 4)/(x − 1) = [x(x − 4)/(x − 1)] · [(x − 1)/(x − 4)] = x (x ≠ 1, 4).

Adding and Subtracting

Find the least common denominator (LCD) by factoring each denominator.

Worked Example 5

3/(x − 2) − 1/(x + 2) = [3(x + 2) − 1(x − 2)] / [(x − 2)(x + 2)] = (3x + 6 − x + 2) / (x^2 − 4) = (2x + 8)/(x^2 − 4).

Factor numerator if possible: 2(x + 4)/(x^2 − 4).

Worked Example 6

2/(x + 1) + x/(x^2 − 1) = 2/(x + 1) + x/[(x − 1)(x + 1)] = [2(x − 1) + x] / (x^2 − 1) = (3x − 2)/(x^2 − 1).

Solving Rational Equations

1. Note domain restrictions.
2. Multiply both sides by the LCD.
3. Solve the resulting polynomial equation.
4. Check each candidate against restrictions—extraneous solutions arise when a value zeros the original denominator.

Worked Example 7

Solve 2/(x − 1) = 3/(x + 2).

Domain: x ≠ 1, −2. LCD: (x − 1)(x + 2).

2(x + 2) = 3(x − 1) → 2x + 4 = 3x − 3 → x = 7. Valid.

Worked Example 8 — Extraneous risk

1/(x − 2) + 1/(x + 2) = 4/(x^2 − 4).

Note x^2 − 4 = (x − 2)(x + 2). Multiply by LCD:

(x + 2) + (x − 2) = 4 → 2x = 4 → x = 2.

But x = 2 zeros the denominator—no solution.

Worked Example 9

Solve (x + 3)/(x − 1) = 2. Domain x ≠ 1.

x + 3 = 2(x − 1) → x + 3 = 2x − 2 → x = 5. Valid.

Complex Fractions

Simplify (1/x + 1/y) by multiplying numerator and denominator by xy: (y + x)/(xy).

Another form: (3/x^2) / (6/x) = (3/x^2) · (x/6) = 1/(2x) for x ≠ 0.

Exam Tips

• State exclusions even if the stem only asks for simplified form.
• After solving, substitute into the original equation.
• Teaching scenarios: if a student cancels x − 2 across a sum, flag illegal cancellation.
• LCD is built from each distinct factor to the highest power appearing—not the product of numerators.

Link to Modeling

Rates and work problems often produce rational equations like 1/x + 1/(x + 2) = 1/3. Clear denominators early to avoid arithmetic errors.

Undefined vs Zero

A rational expression equals zero when the numerator is zero (and denominator nonzero). (x − 2)/(x + 1) = 0 when x = 2; x = −1 is excluded.

Proportions as Rational Equations

a/b = c/d is a rational equation. Cross-multiplication is multiplying by bd. Use this for similar figures: if 4/6 = x/15, then x = 10.

Asymptote Preview

Vertical asymptotes occur where simplified denominators are zero. For y = 1/(x − 3), x = 3 is excluded. This connects rational expressions to graphing rational functions in the functions domain.

Common Denominator Pitfall

Students sometimes add denominators instead of finding LCD: 1/x + 1/y is not 2/(x + y). Correct sum uses xy in the denominator: (y + x)/(xy).

Simplifying Before Operating

Before adding rational expressions, factor every denominator. Skipping this step forces oversized LCDs and arithmetic mistakes under exam timing.

Worked Example 10

Simplify (x^2 − 1)/(x + 1) for x ≠ −1: (x − 1)(x + 1)/(x + 1) = x − 1.

Teaching Misconception: Cross Canceling in Sums

Students may try to cancel x in (x + 2)/x + 1/x. Only factors cancel across multiplication—not terms in a sum. Correct: (x + 2 + 1)/x = (x + 3)/x.

Rate Equations as Rational Models

If a printer completes a job alone in x minutes, its rate is 1/x jobs per minute. Combined with a second printer, rates add when they work simultaneously—setting up 1/x + 1/(x + 10) = 1/12 is a standard Praxis modeling pattern that reduces to a rational equation after clearing denominators.