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Solving Rational Equations using Cross-Multiplication

Solve equations that are fractions on both sides

Atoms Practice
Practice Solving Rational Equations using Cross-Multiplication
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Solving Rational Equations using Cross-Multiplication

A scale model of a racecar is in the ratio of 1:x to the real racecar. The length of the model is 2x21 units, and the length of the real racecar is x2 units. What is the value of x?


A rational equation is an equation where there are rational expressions on both sides of the equal sign. One way to solve rational equations is to use cross-multiplication. Here is an example of a proportion that we can solve using cross-multiplication.

If you need more of a review of cross-multiplication, see the Proportion Properties concept. Otherwise, we will start solving rational equations using cross-multiplication.

Example A

Solve x2x3=3xx+11.

Solution: Use cross-multiplication to solve the problem. You can use the example above as a guideline.

Check your answers. It is possible to get extraneous solutions with rational expressions.

0203030=300+11=011 =04243=344+1145=121545=45

Example B

Solve x+14=3x3.

Solution: Cross-multiply and solve.

x+141200x=3x3=x22x3=x22x15=(x5)(x+3)=5 and 3

Check your answers.

5+14=35364=32 and


Example C

Solve x22x5=x+82.

Solution: Cross-multiply.


Check the answer: (4011)280115=4011+821600121÷2511=12811÷26411=12822

Intro Problem Revisit We need to set up a rational equation and solve for x.


Now cross-multiply.


However, x is a ratio so it must be greater than 0. Therefore x equals 21 and the model is in the ratio 1:21 to the real racecar.

Guided Practice

Solve the following rational equations.

1. xx1=x83

2. x21x+2=2x12

3. 9xx2=43x



xx1x29x+8x26x+8(x4)(x2)x=x83=3x=0=0=4 and 2






9xx24x2x2+27xx(x+27)x=43x=27x+3x2=0=0=0 and 27

Check:x=09002und=43(0)=undx=279+27(27)2=43(27)36729=481 481=481

x=0 is not actually a solution because it is a vertical asymptote for each rational expression, if graphed. Because zero is not part of the domain, it cannot be a solution, and is extraneous.

Problem Set

  1. Is x=2 a solution to x1x4=x21x+4?

Solve the following rational equations.

  1. 2xx+3=8x
  2. 4x+1=x+23
  3. x2x+2=x+32
  4. 3x2x1=2x+1x
  5. x+2x3=x3x2
  6. x+33=2x+6x3
  7. 2x+5x1=2x4
  8. 6x14x2=32x+5
  9. 5x2+110=x382x
  10. x24x+4=2x13

Determine the values of a that make each statement true. If there no values, write none.

  1. 1xa=xx+a, such that there is no solution.
  2. 1xa=xxa, such that there is no solution.
  3. xax=1x+a, such that there is one solution.
  4. 1x+a=xxa, such that there are two integer solutions.


Rational Equation

Rational Equation

A rational equation is an equation that contains a rational expression.

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