7.3 Rational Expressions and Equations

7.3 Rational Expressions and Equations

A rational expression is a ratio of polynomials. A rational equation sets two such ratios, or a ratio and a number, equal. Table 11 groups rationals with radicals because both families have forbidden inputs: even-root radicands below zero, and denominators equal to zero. Skills Insight band 250–262 asks you to rewrite rational expressions. Band 263–275 specifically includes adding and subtracting rational expressions with different denominators. Solving — multiply by an LCD, exclude zeros of denominators, check — is the equation skill that sits on top of that rewriting.

/practice/accuplacer-advanced-algebraPractice questions with detailed explanations

Simplifying a rational expression

Factor, then cancel common factors, never common terms.

Worked: (x^2 − 9) / (x^2 − x − 6)

Numerator: difference of squares (x − 3)(x + 3).

Denominator: (x − 3)(x + 2).

For x ≠ 3, cancel x − 3:

(x + 3)/(x + 2), still with x ≠ 3 and x ≠ −2.

The simplified formula is not defined at x = −2 (original denominator zero) and is not equal to the original at x = 3 (a hole in the graph). Canceling does not restore those x-values to the domain.

Trap: canceling the x^2 terms to get −9 / (−x − 6). You may only cancel factors of the entire numerator and the entire denominator. Another trap cancels (x − 3) and then drops the exclusion x ≠ 3 as if the expressions were identical for every real x.

That last row is not “cancel one x from x^2 only.” Factor first: x(x + 5)/x, then cancel the factor x.

Multiply and divide

Multiply: factor, cancel across a product, multiply remaining factors.

(x/4) · (8/(x + 1)) = (x · 8) / (4(x + 1)) = (2x)/(x + 1) for x ≠ −1.

Divide: multiply by the reciprocal.

(x/4) ÷ (8/(x + 1)) = (x/4) · ((x + 1)/8) = x(x + 1)/32 for x ≠ −1 (and the original divisor ≠ 0, which is the same restriction here).

Keep the reciprocal straight. Dividing by 8/(x + 1) is not dividing by 8 and then by x + 1 as separate steps that invert only the 8. A related trap multiplies by 8/(x + 1) instead of its reciprocal and lands on the product 2x/(x + 1) when the stem asked for a quotient.

Complex fractions such as (1/x) / (1/4) are just division: (1/x) · 4/1 = 4/x for x ≠ 0. Rewriting those is part of band 250–262.

Adding and subtracting unlike denominators

This is the 263–275 skill. Find an LCD, rewrite each fraction, then add numerators. Do not add numerators and add denominators.

Worked: 2/x + 3/(x + 1)

LCD: x(x + 1) (the linear factors are already distinct).

2/x = 2(x + 1) / [x(x + 1)] = (2x + 2)/[x(x + 1)].

3/(x + 1) = 3x / [x(x + 1)].

Sum: (2x + 2 + 3x) / [x(x + 1)] = (5x + 2) / [x(x + 1)].

Exclude x = 0 and x = −1. The numerator 5x + 2 does not factor with the denominator, so this is fully simplified.

If the denominators share a factor, the LCD uses each factor at the highest power that appears. 1/(x^2 − 1) + 2/(x − 1) has LCD (x − 1)(x + 1) because x^2 − 1 already supplies both.

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

Worked subtraction: 4/(x − 2) − 1/(x + 2)

LCD (x − 2)(x + 2) = x^2 − 4.

[4(x + 2) − 1(x − 2)] / [(x − 2)(x + 2)] = (4x + 8 − x + 2) / [(x − 2)(x + 2)] = (3x + 10)/(x^2 − 4).

Watch the minus sign: −(x − 2) is −x + 2, not −x − 2. That single distribution error is enough to miss a 263–275 item.

The false shortcut 2/x + 3/(x + 1) = 5/(2x + 1) adds across. It is never valid unless the fractions were already like terms with a common denominator of 1, which these are not.

Solving rational equations

Clear denominators by multiplying every term by the LCD. Then solve the polynomial equation that remains. Exclude any value that makes an original denominator zero. Check surviving candidates in the original.

Worked example: 3/(x − 1) = 2/(x + 4)

Neither denominator may be zero: x ≠ 1 and x ≠ −4.

Cross-multiplication is LCD multiplication when each side is a single fraction: 3(x + 4) = 2(x − 1).

3x + 12 = 2x − 2.

x = −14.

Check: −14 ≠ 1 and −14 ≠ −4. Left: 3/(−15) = −1/5. Right: 2/(−10) = −1/5. x = −14.

A sign trap: 3(x + 4) = 2(x − 1) is not 3x + 4 = 2x − 1. Distribute through the parentheses. Another trap solves 3(x − 1) = 2(x + 4) by attaching each denominator to the wrong numerator, producing x = −11.

Worked example with an extraneous value: x/(x − 1) = 2 + 1/(x − 1)

Exclude x = 1 from the start.

LCD is x − 1. Multiply through:

x = 2(x − 1) + 1.

x = 2x − 2 + 1.

x = 2x − 1.

x = 1.

But x = 1 was excluded. It makes both original denominators zero. No real solution. The candidate is extraneous.

This is the rational analog of the radical check in Solving Radical Equations and Extraneous Roots. Multiplying by x − 1 is not valid at x = 1, so the algebra of the cleared equation is allowed to produce a number that the original forbids.

Two-denominator solve that keeps both roots: 3/x + 2/(x − 2) = 1

Exclude x = 0 and x = 2. LCD x(x − 2):

3(x − 2) + 2x = x(x − 2).

3x − 6 + 2x = x^2 − 2x.

5x − 6 = x^2 − 2x.

0 = x^2 − 7x + 6 = (x − 6)(x − 1).

Candidates x = 6 and x = 1, neither excluded.

Check x = 6: 3/6 + 2/4 = 1/2 + 1/2 = 1. Keep.

Check x = 1: 3/1 + 2/(−1) = 3 − 2 = 1. Keep.

Both work. As with two-radical equations, a quadratic after clearing denominators is not a signal that one root must die. The exclusions and the original equation decide.

Creating a rational equation

Table 11 includes creating rational equations, not only solving them. “Three over one less than a number equals two over four more than the number” is 3/(x − 1) = 2/(x + 4) — the worked item above, so you still solve rather than guess. Combined rates also produce rationals: two pumps filling a tank in x and x + 4 hours give 1/x + 1/(x + 4) = 1/t. AAF usually keeps the algebra small: LCD, linear or quadratic, then exclusions.

What “undefined” means on AAF

A rational expression is undefined at zeros of the denominator, even after you cancel a factor (hole) or even if a graph has a vertical asymptote there. When an item asks for the domain of (x^2 − 9)/(x^2 − x − 6), the answer is all reals except x = 3 and x = −2, not “except x = −2 only.” Canceling hides a hole; it does not enlarge the original domain.

Handheld calculators are not allowed except with an approved accommodation (College Board calculator policy). Some items show an on-screen calculator; exact simplified rationals such as (5x + 2)/(x(x + 1)) are the usual target, not a rounded decimal. If a calculator icon does appear, use it to check a candidate such as x = −14 in the original fractions, not to skip the LCD.