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Rational Expressions and Equations

Fractions with x in them: when they are undefined, how to simplify, multiply, divide, add and subtract them, how to solve equations with them, and what their graphs look like.

A rational expression is a fraction whose top and bottom are polynomials, like x + 3x − 2. Everything you know about number fractions still works: you simplify by cancelling common factors, and you add with a common denominator. The new rule is that the denominator can never be 0, so some values of x are not allowed. This unit leans on factoring from Polynomials and on the reciprocal graph from Functions and Their Graphs.

Start: excluded valuesJump to equationsJump to practice
Graph of y = 2/(x − 1) + 1 in teal. It has two branches. Dashed gold lines show its asymptotes: the vertical line x = 1 and the horizontal line y = 1. The branches get closer and closer to these lines but never touch them. The point (2, 3), marked in purple, is on the upper right branch.xy−5−4−3−2−1234567−5−4−3−2−1234567y = 2/(x − 1) + 1(2, 3)
The graph of y = 2x − 1 + 1 has a vertical asymptote at x = 1, where the denominator is 0, and a horizontal asymptote at y = 1.

Excluded values

Division by 0 has no answer, so any x that makes a denominator 0 is excluded. To find excluded values, set the denominator equal to 0 and solve. The numerator does not matter: 05 = 0 is fine.

Example 1: Excluded values of a simple fraction

x − 3 = 0Set the denominator equal to 0x = 3So x ≠ 3

Example 2: A denominator that factors

x2 − 5x − 6 = 0Set the denominator equal to 0(x − 6)(x + 1) = 0Factor: −6 and 1 multiply to −6 and add to −5x = 6 or x = −1So x ≠ 6 and x ≠ −1

Simplifying

To simplify, factor the top and the bottom completely, then cancel factors that are the same. You may only cancel factors, things multiplied, never terms that are added. Write the excluded values of the original expression next to your answer, because the simplified form hides them.

On a graph, a value that cancels leaves a hole, a single missing point, while a factor left in the denominator gives a vertical asymptote.

Example 3: Cancel a common factor

(x + 3)(x − 3)x + 3Difference of squares on topx − 3Cancel x + 3; x ≠ −3

Example 4: Factor both parts

(x + 2)(x + 3)(x + 2)(x − 2)Factor the top and the bottomx + 3x − 2Cancel x + 2; x ≠ −2 and x ≠ 2

Example 5: Opposite factors

5 − xx − 5Start: the top and the bottom are opposites−x − 5x − 5Factor −1 out of the top: 5 − x = −(x − 5)−1Cancel x − 5; x ≠ 5

Multiplying and dividing

Multiply rational expressions like number fractions: top times top over bottom times bottom. Factor first, so you can cancel before multiplying. To divide, multiply by the reciprocal of the second fraction: flip it, then multiply.

Example 6: Multiply

(x + 1)(x − 1)3x · (6x2)x + 1Factor x2 − 1(x + 1)(x − 1) · 6x23x(x + 1)Multiply across2x(x − 1)Cancel x + 1 and 3x; 6x23x = 2x. Keep x ≠ 0 and x ≠ −1

Example 7: Divide

x + 4x − 2 · (x + 1)x + 4Divide x + 4x − 2 by x + 4x + 1: flip the second and multiplyx + 1x − 2Cancel x + 4. Excluded: 2, −4 and −1, since we divided by x + 4x + 1

Adding and subtracting

To add or subtract, you need a common denominator. The least common denominator (LCD) is the product of every different factor, each used as many times as it appears most in any one denominator. Rewrite each fraction over the LCD, then add or subtract the tops. When subtracting, put brackets around the whole second top so the minus reaches every term.

Example 8: Same denominator

2x + 1 − (x + 4)x − 3Subtract the tops; brackets on the secondx − 3x − 32x + 1 − x − 4 = x − 31With x ≠ 3

Example 9: Different denominators

3(x + 1) + 2xx(x + 1)The LCD is x(x + 1)5x + 3x(x + 1)3x + 3 + 2x = 5x + 3

Example 10: Factor to find the LCD

x(x + 2)(x − 2) − 1x − 2Factor x2 − 4; the LCD is (x + 2)(x − 2)x − (x + 2)(x + 2)(x − 2)Multiply the second top by x + 2−2(x + 2)(x − 2)x − x − 2 = −2

Complex fractions

A complex fraction has fractions inside its top or bottom. The quickest way to simplify one is to multiply the top and the bottom by the LCD of all the small fractions. That clears them in one move.

Example 11: Clear the small fractions

x(1 + 1x)x(1 − 1x)The LCD of the small fractions is xx + 1x − 1Distribute. Still x ≠ 0, from the original, and x ≠ 1

Example 12: Two different small denominators

2x(1x − 12)2x(x − 2)The LCD is 2x2 − x2x(x − 2)Distribute on top−12x2 − x = −(x − 2); cancel. Keep x ≠ 0 and x ≠ 2

Rational equations and extraneous solutions

To solve an equation with rational expressions: (1) write down the excluded values; (2) multiply both sides by the LCD to clear every fraction; (3) solve the equation that is left; (4) throw out any answer that is an excluded value. Such an answer is called extraneous: it solves the cleared equation but not the original, because the original is undefined there.

Example 13: One solution

x ≠ 0Excluded value6 + x = 4xMultiply both sides by the LCD, 2x6 = 3xx = 2Not excluded, so it stands

Check: 32 + 12 = 2 ✓

Example 14: An extraneous solution

x ≠ 3Excluded valuex = 3 + 2(x − 3)Multiply both sides by x − 3x = 2x − 3x = 3But x ≠ 3, so it is extraneous: no solution

Example 15: Keep one, throw one out

x ≠ 4Excluded valuex2 = 16Multiply both sides by x − 4x = 4 or x = −4x = −4x = 4 is extraneous

Check x = −4 in x2x − 4 = 16x − 4: both sides are 16−8 = −2 ✓

Graphs of y = a/(x − h) + k

The simplest rational function is the reciprocal parent y = 1x. It has two branches and two asymptotes, lines the graph gets closer to but never reaches: x = 0, because 10 is undefined, and y = 0, because 1x is never 0.

Graph of the parent y = 1/x in teal, with its asymptotes, the y-axis x = 0 and the x-axis y = 0, drawn dashed gold. One branch is in the top-right quadrant through (1, 1), marked in purple; the other is in the bottom-left quadrant.xy12345123450y = 1/x(1, 1)
The parent y = 1x with asymptotes x = 0 and y = 0.

The transformation rules from the Functions unit work here too. In y = ax − h + k the graph moves right h and up k, so the vertical asymptote is x = h and the horizontal asymptote is y = k. The number a stretches the branches, and a negative a flips them to the other two corners.

Graph of y = −3/(x + 2) − 1 in teal. Its asymptotes, dashed gold, are x = −2 and y = −1, and they cross at (−2, −1). Because a = −3 is negative, the branches sit in the top-left and bottom-right of that crossing. The point (−1, −4), marked in purple, is one step right of the vertical asymptote.xy−5−4−3−11234−7−6−5−2123450y = −3/(x + 2) − 1(−1, −4)
y = −3x + 2 − 1: h = −2 and k = −1, so the asymptotes are x = −2 and y = −1.

Example 16: Read the asymptotes

x = 5Vertical: where x − 5 = 0y = 2Horizontal: the number added, k(6, 6)One step right: 41 + 2 = 6

Common mistakes

✗Cancelling terms: x + 5x + 3 = 5/3.✓Cancel only factors. x + 3 is not a factor of x + 5.
✗Once x2 − 9x + 3 becomes x − 3, no values are excluded.✓The original still needs x ≠ −3.
✗Adding straight across: 1x + 12 = 2/(x + 2).✓Add the fractions over a common denominator: 1x + 12 = 2 + x2x.
✗Subtracting x + 4x − 3 as −x + 4.✓Bracket the whole second top: −(x + 4) = −x − 4.
✗Keeping every answer to a rational equation.✓Check each one against the excluded values; throw out the extraneous ones.

Practice problems

Printable worksheet (PDF)

Try each one, then open the answer to check your work.

  1. Find the excluded values of x + 1x2 − 16.
    Show answer
    x2 − 16 = 0Set the denominator to 0x = 4 or x = −4
  2. Simplify x2 − 2x − 15x − 5.
    Show answer
    (x − 5)(x + 3)x − 5Factor the topx + 3x ≠ 5
  3. Simplify 4x + 8x2 + 2x.
    Show answer
    4(x + 2)x(x + 2)Factor both4xx ≠ 0 and x ≠ −2
  4. Multiply: x + 2x − 1 · (x2 − 1)x + 2.
    Show answer
    (x + 2)(x + 1)(x − 1)(x − 1)(x + 2)Factor and multiply acrossx + 1Cancel; keep x ≠ 1 and x ≠ −2
  5. Divide x2 − 252x by x + 54x.
    Show answer
    x2 − 252x · (4x)x + 5Flip the second and multiply(x + 5)(x − 5) · 4x2x(x + 5)Factor2(x − 5)Cancel; keep x ≠ 0 and x ≠ −5
  6. Add: 2x + 3 + 1x − 1.
    Show answer
    2(x − 1) + (x + 3)(x + 3)(x − 1)LCD (x + 3)(x − 1)3x + 1(x + 3)(x − 1)2x − 2 + x + 3 = 3x + 1
  7. Subtract: 5x − 2x + 4.
    Show answer
    5(x + 4) − 2xx(x + 4)LCD x(x + 4)3x + 20x(x + 4)5x + 20 − 2x = 3x + 20
  8. Simplify 1 − 4x1 − 16x2.
    Show answer
    x2 − 4xx2 − 16Multiply top and bottom by x2x(x − 4)(x + 4)(x − 4)Factorxx + 4Cancel; keep x ≠ 0, x ≠ 4 and x ≠ −4
  9. Solve 4x = 6x + 1.
    Show answer
    4(x + 1) = 6xCross-multiply; x ≠ 0 and x ≠ −14 = 2xx = 2
  10. Solve 2xx − 1 = 2x − 1 + 3.
    Show answer
    2x = 2 + 3(x − 1)Multiply by x − 1; x ≠ 12x = 3x − 1x = 1Excluded, so extraneous: no solution
  11. Give the asymptotes of y = −1x + 3 + 4.
    Show answer
    x = −3Where x + 3 = 0y = 4The number added at the end

History: the fraction bar

Writing one number above another to make a fraction is old. Jeff Miller’s notes on the earliest uses of fraction symbols, hosted by MacTutor, say Indian mathematicians such as Brahmagupta (around 628) wrote fractions that way but without a bar. Several sources credit the horizontal bar to the Arabic-speaking mathematician al-Hassar around 1200, though the same notes mention much older Egyptian papyri with a similar line, so the first use is uncertain.

According to the same notes, Fibonacci was the first European mathematician to use the bar as we do today, and he called it the virga. Early printers often left the bar out because it was hard to typeset, which is one reason the slanted slash, as in 3/4, became common. That slash is how you type the fractions in this unit’s practice generator.

Connections

Simplifying a rational expression is mostly factoring, from Polynomials: the difference of squares and trinomials show up in almost every problem. Excluded values are the domain idea from Functions and Their Graphs, and the graph of y = ax − h + k is the reciprocal parent moved by the same transformation rules.

Solving a rational equation ends in a linear or quadratic equation, from Linear Equations and Quadratics. Checking for extraneous solutions is the same habit you use with square-root equations in Exponents and Radicals.

Practice generator

Pick a level, or Mixed for a bit of everything. Type fractions with a slash and brackets, like (x + 2)/(x − 3), and exponents with a caret, like x^2. Any equal form of a fully simplified answer is accepted. For a sum or difference, give one fraction. For an equation, type the value, like x = 4.

Sources

Credits: all diagrams on this page are original SVG drawings made for Jugami. Every answer was double-checked.