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# Solving Rational Equations using Cross-Multiplication

## Solve equations that are fractions on both sides

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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\begin{align*}2x-21\end{align*} units, and the length of the real racecar is x2\begin{align*}x^2\end{align*} units. What is the value of x?

### Guidance

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\begin{align*}\frac{x}{2x-3}=\frac{3x}{x+11}\end{align*}.

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\begin{align*}\frac{x+1}{4}=\frac{3}{x-3}\end{align*}.

Solution: Cross-multiply and solve.

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

5+14=35364=32\begin{align*}\frac{5+1}{4}=\frac{3}{5-3} \rightarrow \frac{6}{4}=\frac{3}{2}\end{align*} and

3+14=33324=36\begin{align*}\frac{-3+1}{4}=\frac{3}{-3-3} \rightarrow \frac{-2}{4}=\frac{3}{-6} \end{align*}

#### Example C

Solve x22x5=x+82\begin{align*}\frac{x^2}{2x-5}=\frac{x+8}{2}\end{align*}.

Solution: Cross-multiply.

x22x52x2+11x4011x4011xx=x+82=2x2=0=40=4011

Check the answer: (4011)280115=4011+821600121÷2511=12811÷26411=12822\begin{align*}\frac{\left(\frac{40}{11}\right)^2}{\frac{80}{11}-5}=\frac{\frac{40}{11}+8}{2} \rightarrow \frac{1600}{121} \div \frac{25}{11}=\frac{128}{11} \div 2 \rightarrow \frac{64}{11}=\frac{128}{22}\end{align*}

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

1x=2x21x2\begin{align*}\frac{1}{x} = \frac{2x-21}{x^2}\end{align*}

Now cross-multiply.

x2=x(2x21)x2=2x221x0=x221x0=x(x21)x=0,21

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\begin{align*}\frac{-x}{x-1}=\frac{x-8}{3}\end{align*}

2. x21x+2=2x12\begin{align*}\frac{x^2-1}{x+2}=\frac{2x-1}{2}\end{align*}

3. 9xx2=43x\begin{align*}\frac{9-x}{x^2}=\frac{4}{3x}\end{align*}

1.

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

Check:x=444143=483=43x=2221=28321=63

2.

x21x+22x2+3x23xx=2x12=2x22=0=0

Check:0210+212=2(0)12=12

3.

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\begin{align*}x = 0\end{align*} 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\begin{align*}x=-2\end{align*} a solution to x1x4=x21x+4\begin{align*}\frac{x-1}{x-4}=\frac{x^2-1}{x+4}\end{align*}?

Solve the following rational equations.

1. 2xx+3=8x\begin{align*}\frac{2x}{x+3}=\frac{8}{x}\end{align*}
2. 4x+1=x+23\begin{align*}\frac{4}{x+1}=\frac{x+2}{3}\end{align*}
3. x2x+2=x+32\begin{align*}\frac{x^2}{x+2}=\frac{x+3}{2}\end{align*}
4. 3x2x1=2x+1x\begin{align*}\frac{3x}{2x-1}=\frac{2x+1}{x}\end{align*}
5. x+2x3=x3x2\begin{align*}\frac{x+2}{x-3}=\frac{x}{3x-2}\end{align*}
6. x+33=2x+6x3\begin{align*}\frac{x+3}{-3}=\frac{2x+6}{x-3}\end{align*}
7. 2x+5x1=2x4\begin{align*}\frac{2x+5}{x-1}=\frac{2}{x-4}\end{align*}
8. 6x14x2=32x+5\begin{align*}\frac{6x-1}{4x^2}=\frac{3}{2x+5}\end{align*}
9. 5x2+110=x382x\begin{align*}\frac{5x^2+1}{10}=\frac{x^3-8}{2x}\end{align*}
10. x24x+4=2x13\begin{align*}\frac{x^2-4}{x+4}=\frac{2x-1}{3}\end{align*}

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

1. 1xa=xx+a\begin{align*}\frac{1}{x-a}=\frac{x}{x+a}\end{align*}, such that there is no solution.
2. 1xa=xxa\begin{align*}\frac{1}{x-a}=\frac{x}{x-a}\end{align*}, such that there is no solution.
3. xax=1x+a\begin{align*}\frac{x-a}{x}=\frac{1}{x+a}\end{align*}, such that there is one solution.
4. 1x+a=xxa\begin{align*}\frac{1}{x+a}=\frac{x}{x-a}\end{align*}, such that there are two integer solutions.

### Vocabulary Language: English

Rational Equation

Rational Equation

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