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Rational equations: how to solve them

What a rational equation is, how to solve rational equations by multiplying by the lowest common denominator, and why every answer must be checked.

2 min readUpdated 2 November 2026By Sumzo team (AI-assisted)Reviewed by Sumzo AI (owner-delegated review)

A rational equation is an equation with at least one fraction whose denominator contains the unknown, such as 3x+12=2\frac{3}{x} + \frac{1}{2} = 2. “Rational” comes from “ratio”: these fractions are ratios of polynomials.

The plan for solving rational equations is always the same: get rid of the fractions by multiplying every term by the lowest common denominator, solve the simpler equation that is left, then check each answer in the original equation.

Step 1: note the values the unknown can’t take

Division by zero is undefined, so any value that makes a denominator zero is not allowed. In x+1x−3=3\frac{x + 1}{x - 3} = 3, xx cannot be 3. Write these restrictions down before you start — they decide which answers survive at the end.

How to solve rational equations

  1. Write down the restrictions (values that make a denominator zero).
  2. Find the lowest common denominator (LCD) of all the fractions.
  3. Multiply every term on both sides by the LCD. The fractions cancel.
  4. Solve the equation that is left — it will be linear or quadratic.
  5. Reject any answer that breaks a restriction, and check the rest in the original equation.

Worked example

Easy: one fraction

Solve 3x=6\frac{3}{x} = 6.

Show the working
  1. Restriction: x≠0x \neq 0. Multiply both sides by xx.
    3=6x3 = 6x
  2. Divide by 6.
    x=12x = \frac{1}{2}
  3. Check: 31/2=6\frac{3}{1/2} = 6.
    3÷12=6✓3 \div \tfrac{1}{2} = 6 \checkmark

Answer: x=12x = \frac{1}{2}

Checked by Sumzo’s math engine

Worked example

Medium: two fractions with different denominators

Solve 3x+12=2\frac{3}{x} + \frac{1}{2} = 2.

Show the working
  1. Restriction: x≠0x \neq 0. The LCD of xx and 2 is 2x2x; multiply every term by it.
    6+x=4x6 + x = 4x
  2. Subtract xx from both sides.
    6=3x6 = 3x
  3. Divide by 3.
    x=2x = 2
  4. Check.
    32+12=2✓\tfrac{3}{2} + \tfrac{1}{2} = 2 \checkmark

Answer: x=2x = 2

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Worked example

Medium: the unknown on top and bottom

Solve x+1x−3=3\frac{x + 1}{x - 3} = 3.

Show the working
  1. Restriction: x≠3x \neq 3. Multiply both sides by x−3x - 3.
    x+1=3(x−3)=3x−9x + 1 = 3(x - 3) = 3x - 9
  2. Subtract xx and add 9.
    10=2x10 = 2x
  3. Divide by 2. Since 5 ≠ 3, the answer is allowed.
    x=5x = 5
  4. Check.
    5+15−3=62=3✓\tfrac{5 + 1}{5 - 3} = \tfrac{6}{2} = 3 \checkmark

Answer: x=5x = 5

Checked by Sumzo’s math engine

Worked example

Hard: an answer that has to be rejected

Solve xx−2+1=2x−2\frac{x}{x - 2} + 1 = \frac{2}{x - 2}.

Show the working
  1. Restriction: x≠2x \neq 2. Multiply every term — including the 1 — by x−2x - 2.
    x+(x−2)=2x + (x - 2) = 2
  2. Simplify and solve.
    2x−2=2⇒x=22x - 2 = 2 \Rightarrow x = 2
  3. But x=2x = 2 makes the denominators zero, so it is an extraneous solution and must be rejected.
    no solution\text{no solution}

Answer: No solution (x=2x = 2 is extraneous)

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Worked example

Hard: a rational equation that becomes a quadratic

Solve 1x+1x+2=34\frac{1}{x} + \frac{1}{x + 2} = \frac{3}{4}.

Show the working
  1. Restrictions: x≠0x \neq 0 and x≠−2x \neq -2. Multiply every term by the LCD, 4x(x+2)4x(x + 2).
    4(x+2)+4x=3x(x+2)4(x + 2) + 4x = 3x(x + 2)
  2. Expand and collect everything on one side.
    3x2−2x−8=03x^2 - 2x - 8 = 0
  3. Factor.
    (3x+4)(x−2)=0(3x + 4)(x - 2) = 0
  4. Both answers are allowed.
    x=2 or x=−43x = 2 \text{ or } x = -\tfrac{4}{3}
  5. Check x=2x = 2: 12+14=34\frac{1}{2} + \frac{1}{4} = \frac{3}{4}. Check x=−43x = -\frac{4}{3}: −34+32=34-\frac{3}{4} + \frac{3}{2} = \frac{3}{4}.
    ✓ ✓\checkmark\ \checkmark

Answer: x=2x = 2 or x=−43x = -\frac{4}{3}

Checked by Sumzo’s math engine

Why check for extraneous solutions?

Multiplying by an expression that contains xx can add solutions that weren’t there before — the new equation is true at those values, but the original isn’t even defined there. That is why the last step of solving rational equations is never optional.

Proportions: the simplest rational equations

A proportion says two ratios are equal, like x4=912\frac{x}{4} = \frac{9}{12}. Because each side is a single fraction, cross-multiplying is allowed here.

Worked example

Solve a proportion

Solve x4=912\frac{x}{4} = \frac{9}{12}.

Show the working
  1. Cross-multiply.
    12x=4×9=3612x = 4 \times 9 = 36
  2. Divide by 12.
    x=3x = 3
  3. Check: 34\frac{3}{4} and 912\frac{9}{12} are equal fractions.
    912=34✓\tfrac{9}{12} = \tfrac{3}{4} \checkmark

Answer: x=3x = 3

Checked by Sumzo’s math engine

Rational equations in real life: working together

Rate problems lead straight to rational equations. If a job takes tt hours, then 1t\frac{1}{t} of it is done each hour, and rates working together add up.

Worked example

Two pipes filling a tank

Pipe A fills a tank in 3 hours and pipe B in 6 hours. How long do they take together?

Show the working
  1. Let tt be the time together. Per hour, A fills 13\frac{1}{3} and B fills 16\frac{1}{6}; together they fill 1t\frac{1}{t}.
    13+16=1t\frac{1}{3} + \frac{1}{6} = \frac{1}{t}
  2. Multiply every term by the LCD, 6t6t.
    2t+t=62t + t = 6
  3. Solve.
    3t=6⇒t=23t = 6 \Rightarrow t = 2
  4. Check: in 2 hours A fills 23\frac{2}{3} and B fills 13\frac{1}{3} — one full tank.
    23+13=1✓\tfrac{2}{3} + \tfrac{1}{3} = 1 \checkmark

Answer: 2 hours

Checked by Sumzo’s math engine

Common mistakes

  • Forgetting the restrictions, then keeping an answer that makes a denominator zero.
  • Multiplying only the fractions by the LCD and forgetting whole-number terms, like the +1+1 in the example whose answer had to be rejected.
  • Cross-multiplying when a side has more than one term.
  • Sign errors when expanding: 3(x−3)3(x - 3) is 3x−93x - 9.

Frequently asked questions

What is a rational equation?

An equation containing a fraction with the unknown in the denominator, such as 3x=6\frac{3}{x} = 6 or x+1x−3=3\frac{x + 1}{x - 3} = 3.

What is an extraneous solution?

An answer that comes out of the working but doesn’t satisfy the original equation — usually because it makes a denominator zero. It must be rejected.

Do rational equations always have a solution?

No. If every answer turns out to be extraneous, the equation has no solution — as in the example above where x=2x = 2 had to be rejected.

When can I cross-multiply?

Only when the equation is one fraction equal to one fraction. Otherwise multiply every term by the lowest common denominator.

Calculators

Practice

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