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Operations with Rational Expressions

Simplify, multiply, divide, add and subtract fractions that contain variables.

Practice this skill 25 min Rational expression operations Leer en español
Key idea

What you'll learn

A rational expression is a fraction with a polynomial on the top and the bottom, such as \dfrac{x + 3}{x - 2} or \dfrac{60}{r}. You already know how to simplify, multiply, divide, add and subtract number fractions. In this lesson you will do exactly the same things with x in the fraction.

You will learn to:

  • Simplify a rational expression by factoring and canceling common factors.
  • State the values that make an expression undefined (excluded values).
  • Multiply and divide rational expressions.
  • Add and subtract them, including when the denominators are different.

Why it matters on the GED. The test asks you to pick the expression that is equivalent to a sum, product or quotient of fractions with variables. The wrong choices come from predictable errors: canceling terms instead of factors, adding the denominators, or forgetting to distribute a minus sign. This lesson trains you to avoid each one.

Real life. Rational expressions describe time and rates. If you drive 60 miles at r miles per hour, the trip takes \dfrac{60}{r} hours. Adding the time there and the time back is adding rational expressions.

The big rule: everything you do with x-fractions mirrors what you do with number fractions. When stuck, ask, "What would I do with \tfrac{2}{3} + \tfrac{1}{4}?"

Words to know

  • Rational expression — a fraction whose numerator and denominator are polynomials, such as \dfrac{x^2 - 9}{x + 1}.
  • Numerator / denominator — the top / the bottom of a fraction.
  • Factor — a piece that is multiplied. In (x - 3)(x + 3), both (x - 3) and (x + 3) are factors.
  • Term — a piece that is added or subtracted. In x + 3, both x and 3 are terms.
  • Simplest form — the numerator and denominator have no common factor except 1.
  • Excluded value (restriction) — a value of x that makes a denominator 0. For \dfrac{5}{x - 4}, x \ne 4.
  • Undefined — has no value because it would mean dividing by zero.
  • Reciprocal — a fraction flipped over. The reciprocal of \dfrac{x + 2}{x} is \dfrac{x}{x + 2}.
  • Least common denominator (LCD) — the smallest expression that every denominator divides into. The LCD of x and x + 1 is x(x + 1).
  • Equivalent expressions — expressions that give the same value for every allowed x.

All Math words in the glossary

Simplifying: cancel factors, never terms

Rule. You may cancel a factor that appears in both the numerator and the denominator. You may not cancel a term.

Number version: \dfrac{12}{18} = \dfrac{6 \cdot 2}{6 \cdot 3} = \dfrac{2}{3}.

Fraction bars: a bar split into 18 parts with 12 shaded is the same length as a bar split into 3 parts with 2 shaded. 12/18 = (6 times 2)/(6 times 3) = 2/3, because the common factor 6 cancels.12/182/312/18 = (6 · 2)/(6 · 3) = 2/3: cancel the common factor 6

Steps to simplify

  1. Factor the numerator and denominator completely (GCF, trinomials, difference of squares).
  2. Write down the excluded values: every x that makes the original denominator 0.
  3. Cancel common factors.

Example. \dfrac{4x + 12}{4} = \dfrac{4(x + 3)}{4} = x + 3. The 4 is a factor of the whole numerator, so it cancels.

Example. \dfrac{x^2 - 9}{x^2 + 5x + 6} = \dfrac{(x - 3)(x + 3)}{(x + 2)(x + 3)} = \dfrac{x - 3}{x + 2}, with x \ne -2 and x \ne -3.

The forbidden cancel. In \dfrac{x + 6}{x} you cannot cancel the x's, because the top x is only a term. Test with x = 2: \dfrac{2 + 6}{2} = 4, but "canceled" it would be 6 or 7. Not equal, so the cancel was wrong.

Factor versus term. Left: in 4(x + 3) over 4, the 4 is a factor of the whole numerator, so the 4s cancel and the result is x + 3. Right: in (x + 6) over x, the top x is only a term added to 6, so it cannot cancel; testing x = 2 gives (2 + 6)/2 = 4, not 6.Factor: can cancel4(x + 3)4= x + 34 multiplies the WHOLE topTerm: cannot cancelx+ 6x≠ 6x = 2: (2 + 6)/2 = 4, not 6

Opposite factors. x - 5 and 5 - x are opposites: 5 - x = -(x - 5). So \dfrac{5 - x}{x - 5} = -1.

Excluded values come from the original. Even after (x + 3) cancels above, x = -3 is still not allowed, because the original expression has a zero denominator there.

Multiplying and dividing

Multiply: multiply straight across, but factor and cancel first so the numbers stay small.

\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}

Example: \dfrac{x^2 - 4}{3x} \cdot \dfrac{6x^2}{x + 2}

  1. Factor: \dfrac{(x - 2)(x + 2)}{3x} \cdot \dfrac{6x^2}{x + 2}.
  2. Cancel (x + 2) top and bottom; cancel 3x into 6x^2 to leave 2x.
  3. Result: 2x(x - 2) = 2x^2 - 4x (with x \ne 0, -2).
Multiplying rational expressions: (x − 2)(x + 2) over 3x, times 6x squared over (x + 2). The factor (x + 2) cancels top and bottom, and 6x squared divided by 3x leaves 2x. The product is 2x(x − 2) = 2x squared − 4x, with x not equal to 0 or −2.(x − 2)(x + 2)3x·6x²(x + 2)cancel (x + 2)6x² ÷ 3x = 2x=2x(x − 2) = 2x² − 4xFactor first, cancel, then multiply what is left (x ≠ 0, −2)

Divide: keep the first fraction, change \div to \times, flip the second fraction. Then multiply.

\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}
Keep, change, flip: a/b divided by c/d becomes a/b times d/c. Keep the first fraction, change the division sign to multiplication, and flip only the second fraction.ab÷cdab×dcKEEPCHANGEFLIPonly the secondfraction flips

Example: \dfrac{x^2 - 4}{x + 3} \div \dfrac{x - 2}{x + 3} = \dfrac{(x - 2)(x + 2)}{x + 3} \cdot \dfrac{x + 3}{x - 2} = x + 2.

Only the second fraction flips. Flipping the first one (or both) is a common wrong answer.

Restrictions for division. The flipped fraction's old numerator becomes a denominator, so it cannot be zero either. Above, x \ne -3 and also x \ne 2.

Adding and subtracting

Same denominator: add or subtract the numerators; keep the denominator.

\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c}

Never add the denominators. \dfrac{1}{x} + \dfrac{1}{x} = \dfrac{2}{x}, not \dfrac{2}{2x}.

Subtracting? Put the second numerator in parentheses. The minus sign applies to every term in it:

\frac{3x + 1}{x + 2} - \frac{x - 5}{x + 2} = \frac{3x + 1 - (x - 5)}{x + 2} = \frac{3x + 1 - x + 5}{x + 2} = \frac{2x + 6}{x + 2}
Subtracting a numerator: in 3x + 1 − (x − 5), arrows show the minus sign applying to both terms in the parentheses, giving 3x + 1 − x + 5 = 2x + 6.3x + 1−(x − 5)3x + 1− x+ 5= 2x + 6the minus sign reachesEVERY term in the group

Different denominators:

  1. Factor each denominator.
  2. Build the LCD: use every different factor, as many times as it appears in any one denominator.
  3. Multiply each numerator by whatever its denominator is missing.
  4. Combine the numerators; simplify; factor and cancel if possible.

Example: \dfrac{3}{x} + \dfrac{2}{x + 1}. LCD = x(x + 1).

\frac{3(x + 1)}{x(x + 1)} + \frac{2x}{x(x + 1)} = \frac{3x + 3 + 2x}{x(x + 1)} = \frac{5x + 3}{x(x + 1)}
DenominatorsLCD
x and 2x2x
x and x + 1x(x + 1)
x - 2 and x^2 - 4(x - 2)(x + 2)
x + 3 and x - 1(x + 3)(x - 1)
Tip

Opposite denominators and answers that look different

Opposite factors. The binomials x - 2 and 2 - x are opposites: 2 - x = -(x - 2). Spotting them saves you from building a huge LCD.

Example: \dfrac{3}{x - 2} + \dfrac{1}{2 - x}. Rewrite the second fraction by moving the negative sign out of its denominator:

\frac{1}{2 - x} = \frac{1}{-(x - 2)} = -\frac{1}{x - 2}

Now the denominators match:

\frac{3}{x - 2} - \frac{1}{x - 2} = \frac{2}{x - 2}

Check with x = 4: \dfrac{3}{2} + \dfrac{1}{-2} = 1, and \dfrac{2}{2} = 1 ✓.

Same answer, different look. The GED may write the correct answer in a form that looks different from yours. All of these are equivalent:

\frac{2x + 6}{x + 2} = \frac{2(x + 3)}{x + 2} \qquad -\frac{1}{x - 2} = \frac{1}{2 - x} = \frac{-1}{x - 2}

If your answer is not among the choices, try factoring it, multiplying it out, or moving a negative sign. If you are still unsure, plug the same test value into your answer and into each choice.

Formula

Fraction rules (memorize — not on the formula sheet)

\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd} \qquad \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c} = \frac{ad}{bc} \frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c} \qquad \frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd} \frac{ak}{bk} = \frac{a}{b} \quad (k \ne 0)
  • a, b, c, d, k can be numbers or whole polynomials. Wrap each one in parentheses when you substitute.
  • None of the denominators may be 0.

These rules are not on the GED formula sheet. They are the same rules you use for number fractions, so memorize them in that form.

Worked example

Example 1 (easy): Simplify and state restrictions

Simplify \dfrac{x^2 - 9}{x^2 + 5x + 6} and state the excluded values.

Example 1: (x − 3)(x + 3) over (x + 2)(x + 3). The common factor (x + 3) is crossed out on top and bottom, leaving (x − 3)/(x + 2). A number line marks the excluded values x = −3 and x = −2 with open red circles, because the original denominator is zero there.(x − 3)(x + 3)(x + 2)(x + 3)=x − 3x + 2−5−4−3−2−1012x ≠ −3x ≠ −2excluded values comefrom the ORIGINAL bottom
  1. Factor the top (difference of squares): x^2 - 9 = (x - 3)(x + 3).
  2. Factor the bottom: x^2 + 5x + 6 = (x + 2)(x + 3).
  3. Excluded values from the original bottom: x \ne -2 and x \ne -3.
  4. Cancel the common factor (x + 3).

Answer: \dfrac{x - 3}{x + 2}, with x \ne -2, -3

Check: x = 1: original \dfrac{1 - 9}{1 + 5 + 6} = \dfrac{-8}{12} = -\dfrac{2}{3}; simplified \dfrac{1 - 3}{1 + 2} = -\dfrac{2}{3} ✓

Worked example

Example 2: Divide

Simplify \dfrac{x^2 - x - 6}{x^2 - 9} \div \dfrac{x + 2}{x + 3}.

Example 2 after keep-change-flip and factoring: (x − 3)(x + 2) over (x − 3)(x + 3), times (x + 3) over (x + 2). Matching factors are colour-coded: (x − 3) with (x − 3), (x + 3) with (x + 3), (x + 2) with (x + 2). All are crossed out, so the quotient is 1 for x not equal to 3, −3 or −2.(x − 3)(x + 2)(x − 3)(x + 3)·(x + 3)(x + 2)= 1Every factor cancels with a matching partner.x ≠ 3, −3, −2
  1. Keep, change, flip: \dfrac{x^2 - x - 6}{x^2 - 9} \cdot \dfrac{x + 3}{x + 2}.
  2. Factor: x^2 - x - 6 = (x - 3)(x + 2) and x^2 - 9 = (x - 3)(x + 3).
  3. Write it out: \dfrac{(x - 3)(x + 2)}{(x - 3)(x + 3)} \cdot \dfrac{x + 3}{x + 2}.
  4. Cancel (x - 3), (x + 2) and (x + 3). Everything cancels.

Answer: 1 (for x \ne 3, -3, -2)

Check: x = 4: the first fraction is \dfrac{16 - 4 - 6}{16 - 9} = \dfrac{6}{7} and the second is \dfrac{4 + 2}{4 + 3} = \dfrac{6}{7}, so the quotient is \dfrac{6}{7} \div \dfrac{6}{7} = 1 ✓

Worked example

Example 3: Add with different denominators

Write \dfrac{3}{x} + \dfrac{2}{x + 1} as a single fraction.

Example 3: building the LCD x(x + 1). 3/x is multiplied by (x + 1)/(x + 1) to get 3(x + 1) over x(x + 1). 2/(x + 1) is multiplied by x/x to get 2x over x(x + 1). Adding the numerators gives (5x + 3) over x(x + 1).3x×(x + 1)(x + 1)=3(x + 1)x(x + 1)2x + 1×xx=2xx(x + 1)5x + 3x(x + 1)LCD = x(x + 1)Multiply each top by what its bottom is missing
  1. LCD = x(x + 1).
  2. First fraction is missing (x + 1): \dfrac{3(x + 1)}{x(x + 1)}.
  3. Second fraction is missing x: \dfrac{2x}{x(x + 1)}.
  4. Add numerators: 3x + 3 + 2x = 5x + 3.

Answer: \dfrac{5x + 3}{x(x + 1)}

Check: x = 2: \dfrac{3}{2} + \dfrac{2}{3} = \dfrac{13}{6}; formula \dfrac{13}{2 \cdot 3} = \dfrac{13}{6} ✓

Worked example

Example 4 (real world): Total driving time

Ana drives 60 miles to a job site at r miles per hour and drives back 10 mph faster. Write her total driving time as a single rational expression.

Example 4 trip diagram: 60 miles to the job site at r miles per hour and 60 miles back at r + 10 miles per hour. Time = distance divided by rate, so total time = 60/r + 60/(r + 10) = (120r + 600) over r(r + 10). With r = 50: 1.2 hours + 1 hour = 2.2 hours.StartJob sitethere: 60 mi at r mphback: 60 mi at (r + 10) mph60r+60r + 10=120r + 600r(r + 10)time = distance ÷ rate. r = 50: 1.2 h + 1 h = 2.2 h
  1. Time = distance ÷ rate. There: \dfrac{60}{r}. Back: \dfrac{60}{r + 10}.
  2. Total: \dfrac{60}{r} + \dfrac{60}{r + 10}. LCD = r(r + 10).
  3. Rewrite: \dfrac{60(r + 10)}{r(r + 10)} + \dfrac{60r}{r(r + 10)}.
  4. Add numerators: 60r + 600 + 60r = 120r + 600.

Answer: \dfrac{120r + 600}{r(r + 10)} hours

Check: r = 50: \dfrac{60}{50} + \dfrac{60}{60} = 1.2 + 1 = 2.2 h; formula \dfrac{6000 + 600}{50 \cdot 60} = \dfrac{6600}{3000} = 2.2 ✓

Worked example

Example 5 (GED-hard, work backwards): Missing side of a rectangle

A rectangle has area \dfrac{x^2 - 9}{x + 1} square feet and length \dfrac{x + 3}{2x + 2} feet. Find an expression for its width.

Example 5 rectangle drawn to scale for x = 5. Its area is (x squared − 9)/(x + 1) square feet and its length is (x + 3)/(2x + 2) feet; the width is area divided by length, 2x − 6. At x = 5 the area is 8/3, the length 2/3 and the width 4 feet.area = (x² − 9)/(x + 1)width = area ÷ length = 2x − 6length = (x + 3)/(2x + 2)Drawn for x = 5: area 8/3 sq ft, length 2/3 ft, width 4 ft(8/3) ÷ (2/3) = 4 = 2(5) − 6
  1. Width = area ÷ length: \dfrac{x^2 - 9}{x + 1} \div \dfrac{x + 3}{2x + 2}.
  2. Flip the second: \dfrac{x^2 - 9}{x + 1} \cdot \dfrac{2x + 2}{x + 3}.
  3. Factor: \dfrac{(x - 3)(x + 3)}{x + 1} \cdot \dfrac{2(x + 1)}{x + 3}.
  4. Cancel (x + 3) and (x + 1): 2(x - 3).

Answer: width = 2(x - 3) = 2x - 6 feet

Check: x = 5: area \dfrac{16}{6} = \dfrac{8}{3}, length \dfrac{8}{12} = \dfrac{2}{3}, width \dfrac{8}{3} \div \dfrac{2}{3} = 4, and 2(5) - 6 = 4 ✓

Calculator tip

Calculator tip: the plug-in check

The TI-30XS MultiView cannot simplify algebra, but it can check whether two expressions are equivalent.

  1. Pick a test value that is not an excluded value, such as x = 4 or x = 5.
  2. Use the n/d key to type stacked fractions. For \dfrac{3}{x} + \dfrac{2}{x + 1} at x = 2: n/d 3 ▼ 2 ▶ + n/d 2 ▼ ( 2 + 1 ) ▶ enter.
  3. Evaluate each answer choice the same way. The correct choice gives the same value.
  4. Press toggle to switch a result between fraction and decimal form, which makes comparing easier.

When not to use it: when the factoring is easy, doing it by hand is faster. And be careful: a single test value can occasionally match a wrong answer by chance, so if two choices match, try a second value.

Common mistake

Common mistakes

  • Canceling terms. Wrong: \dfrac{x + 6}{x} = 6. Right: it cannot be simplified. Only cancel factors of the entire top and bottom.
  • Adding denominators. Wrong: \dfrac{2}{x} + \dfrac{5}{x} = \dfrac{7}{2x}. Right: \dfrac{7}{x}. Denominators stay the same when they match.
  • Not distributing the minus. Wrong: \dfrac{3x + 1 - x - 5}{x + 2}. Right: \dfrac{3x + 1 - x + 5}{x + 2}, because -(x - 5) = -x + 5.
  • Flipping the wrong fraction. When dividing, flip only the second fraction.
  • Forgetting to multiply the numerator. When building the LCD, whatever you multiply the bottom by, multiply the top by too.
  • Losing restrictions. Excluded values come from the original denominators (and, for division, the numerator of the divisor), even if those factors cancel.
Test strategy

GED test strategy

  • Factor everything first. Most rational-expression questions turn easy the moment every top and bottom is factored.
  • Plug in a number (not 0, 1 or an excluded value) to compare the original with each answer choice.
  • Eliminate by form. If a choice adds the denominators or "cancels" a single term, cross it off.
  • Watch the subtraction sign — draw parentheses around the second numerator before you combine.
  • Read the question twice: "simplify," "equivalent to," and "excluded values" are different tasks.
  • Time: if a choice has not matched after one test value, move to the next choice rather than re-doing algebra.

Check yourself

Rational expression operationsNo calculator

Which expression is equivalent to \dfrac{4x + 12}{4}?

Answer choices
Hint

Can you factor the numerator so that a 4 is multiplied by the whole thing?

Check yourself

Rational expression operationsNo calculator

Simplify: \dfrac{x^2 - 25}{x - 5}, for x \ne 5.

Answer choices
Hint

Factor the numerator. It is a difference of two squares.

Summary

  • Simplify by factoring completely and canceling common factors, never terms.
  • Excluded values make an original denominator zero.
  • Multiply straight across; to divide, flip the second fraction and multiply.
  • Add or subtract only with a common denominator; never add denominators; put parentheses around a subtracted numerator.
  • Check any answer by plugging in a safe test value.
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