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Mathematics LibreTexts

8.7: Rational Equations

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Rational Equations

Rational Equations

When one rational expression is set equal to another rational expression, a rational equation results.

Some examples of rational equations are the following (except for number 5):

Example 8.7.1

3x4=152

Example 8.7.2

x+1x2=x7x3

Example 8.7.3

5a2=10

Example 8.7.4

3x+x3x+1=65x

Example 8.7.5

x6x+1 is a rational expression, not a rational equation.

The Logic Behind The Process

It seems most reasonable that an equation without any fractions would be easier to solve than an equation with fractions. Our goal, then, is to convert any rational equation to an equation that contains no fractions. This is easily done.

To develop this method, let’s consider the rational equation

16+x4=1712

The LCD is 12. We know that we can multiply both sides of an equation by the same nonzero quantity, so we’ll multiply both sides by the LCD, 12.

12(16+x4)=121712

Now distribute 12 to each term on the left side using the distributive property.

1216+12x4=121712

Now divide to eliminate all denominators.

21+3x=172+3x=17

Now there are no more fractions, and we can solve this equation using our previous techniques to obtain 5 as the solution.

The Process

We have cleared the equation of fractions by multiplying both sides by the LCD. This development generates the following rule.

Clearing an Equation of Fractions

To clear an equation of fractions, multiply both sides of the equation by the LCD.

When multiplying both sides of the equation by the LCD, we use the distributive property to distribute the LCD to each term. This means we can simplify the above rule.

Clearing an Equation of Fractions

To clear an equation of fractions, multiply every term on both sides of the equation by the LCD.

The complete method for solving a rational equation is

1. Determine all the values that must be excluded from consideration by finding the values that will produce zero in the denominator (and thus, division by zero). These excluded values are not in the domain of the equation and are called nondomain values.

2. Clear the equation of fractions by multiplying every term by the LCD.

3. Solve this nonfractional equation for the variable. Check to see if any of these potential solutions are excluded values.

4. Check the solution by substitution.

Extraneous Solutions

Extraneous Solutions

Potential solutions that have been excluded because they make an expression undefined (or produce a false statement for an equation) are called extraneous solutions. Extraneous solutions are discarded. If there are no other potential solutions, the equation has no solution.

Sample Set A

Solve the following rational equations.

Example 8.7.6

3x4=152 Since the denominators are constants, there are no excluded values. No values must be excluded. The LCD is 4. Multiply each term by 443x4=415243x4=241523x=2153x=30x=1010 is not an excluded value. Check it as a solution.

Check:

3x4=1523(10)4=152 Is this correct? 304=152 Is this correct? 152=152 Yes, this is correct 

Example 8.7.7

4x1=2x+61 and 6 are nondomain values. Exclude them from the solutionThe LCD is (x1)(x+6) Multiply every term by the LCD (x1)(x+6)4x1=(x1)(x+6)2x+6(x1)(x+6)4x1=(x1)(x+6)2x+64(x+6)=2(x1) Solve this nonfractional equation 4x+24=2x22x=26x=1313 is not an excluded value. Check it as a solution

Check:

4x1=2x+64131=213+6 Is this correct?414=27 Is this correct?27=213+6 Yes, this is correct 

13 is the solution.

Example 8.7.8

4aa4=2+16a4.4 is a nondomain value. Exclude it from consideration The LCD is a4. Multiply every term by a4(a4)4aa4=2(a4)+(a4)16a4(a4)4aa4=2(a4)+(a4)16a44a=2(a4)+16 Solve this nonfractional equation 4a=2a8+164a=2a+82a=8a=4

This value, a=4, has been excluded from consideration. It is not to be considered as a solution. It is extraneous. As there are no other potential solutions to consider, we conclude that this equation has no solution.

Practice Set A

Solve the following rational equations.

Practice Problem 8.7.1

2x5=x146

Answer

x=10

Practice Problem 8.7.2

\boldsymbol{\dfrac{3a}{a-1} = \dfrac{3a + 8}}

Answer

a=2

Practice Problem 8.7.3

3y3+2=yy3

Answer

y=3 is extraneous, so no solution.

Sample Set B

Solve the following rational equations.

Example 8.7.9

3x+4xx1=4x2+x+5x2x Factor all denominators to find any excluded values and the LCD  Nondomain values are 0 and 1. Exclude them from consideration. 3x+4xx1=4x2+x+5x(x1) The LCD is x(x1). Multiply each term by x(x1) and simplify 
x(x1)3x+x(x1)4xx1=x(x1)4x2+x+5x(x1).
3(x1)+4xx=4x2+x+5 Solve this nonfractional equation to obtain the potential solutions 3x3+4x2=4x2+x+53x3=x+52x=8x=44 is not an excluded value. Check it as a solution 

Check:

3x+4xx1=4x2+x+5x2x34+4441=442+4+5164 Is this correct? 34+163=64+4+512 Is this correct? 912+6412=7312 Is this correct? 7312=7312 Yes, this is correct 

4 is the solution.

The zero-factor property can be used to solve certain types of rational equations. We studied the zero-factor property in Section 5.1, and you may remember that it states that if a and b are real numbers and that ab=0, then either or both a=0 or b=0.The zero-factor property is useful in solving the following rational equation.

Example 8.7.10

3a22a=1 Zero is an excluded value.  The LCD is a2 Multiply each term by a2 and simplify a23a2a22a=1a232a=a2 Solve this nonfractional quadratic equation. Set it equal to zero 0=a2+2a30=(a+3)(a1)a=3,a=1 Check these as solutions 

Check:

If a=3:3(3)223=1 Is this correct? 39+23=1 Is this correct? 13+23=1 Is this correct? 1=1 Yes, this is correct a=3 Checks and is a solution If a=1:3(1)221=1 Is this correct? 3121=1 Is this correct? 1=1 Yes, this is correct. a=1 Checks and is a solution 

3 and 1 are the solutions.

Practice Set B

Practice Problem 8.7.4

Solve the equation a+3a2=a+1a1

Answer

a=13

Practice Problem 8.7.5

Solve the equation 1x11x+1=2xx21

Answer

This equation has no solution. x=1 is extraneous.

Section 7.6 Exercises

For the following problems, solve the rational equations.

Exercise 8.7.1

32x=163

Answer

x=6

Exercise 8.7.2

54y=274

Exercise 8.7.3

8y=23

Answer

y=12

Exercise 8.7.4

x28=37

Exercise 8.7.5

x+14=x32

Answer

x=7

Exercise 8.7.6

a+36=a14

Exercise 8.7.7

y36=y+14

Answer

y=9

Exercise 8.7.8

x78=x+56

Exercise 8.7.9

a+69a16=0

Answer

a=15

Exercise 8.7.10

y+114=y+810

Exercise 8.7.11

b+12+6=b43

Answer

b=47

Exercise 8.7.12

m+32+1=m45

Exercise 8.7.13

a62+4=1

Answer

a=4

Exercise 8.7.14

b+113+8=6

Exercise 8.7.15

y1y+2=y+3y2

Answer

y=12

Exercise 8.7.16

x+2x6=x1x+2

Exercise 8.7.17

3m+12m=43

Answer

m=3

Exercise 8.7.18

2k+73k=54

Exercise 8.7.19

4x+2=1

Answer

x=2

Exercise 8.7.20

6x3=1

Exercise 8.7.21

a3+10+a4=6

Answer

a=6

Exercise 8.7.22

k+175k2=2k

Exercise 8.7.23

2b+13b5=14

Answer

b=95

Exercise 8.7.24

3a+42a7=79

Exercise 8.7.25

xx+3xx2=10x2+x6

Answer

x=2

Exercise 8.7.26

3yy1+2yy6=5y215y+20y27y+6

Exercise 8.7.27

4aa+23aa1=a28a4a2+a2

Answer

a=2

Exercise 8.7.28

3a7a3=4a10a3

Exercise 8.7.29

2x5x6=x+1x6

Answer

No solution; 6 is an excluded value.

Exercise 8.7.30

3x+4+5x+4=3x1

Exercise 8.7.31

2y+2+8y+2=9y+3

Answer

y=12

Exercise 8.7.32

4a2+2a=3a2+a2

Exercise 8.7.33

2b(b+2)=3b2+6b+8

Answer

b=8

Exercise 8.7.34

xx1+3xx4=4x28x+1x25x+4

Exercise 8.7.35

4xx+2xx+1=3x2+4x+4x2+3x+2

Answer

no solution

Exercise 8.7.36

2a54a2a26a+5=3a1

Exercise 8.7.37

1x+42x+1=4x+19x2+5x+4

Answer

No solution;  4 is an excluded value.

Exercise 8.7.38

2x2+1x=1

Exercise 8.7.39

6y25y=1

Answer

y=6,1

Exercise 8.7.40

12a24a=1

Exercise 8.7.41

20x21x=1

Answer

x=4,5

Exercise 8.7.42

12y+12y2=3

Exercise 8.7.43

16b2+12b=4

Answer

y=4,1

Exercise 8.7.44

1x2=1

Exercise 8.7.45

16y2=1

Answer

y=4,4

Exercise 8.7.46

25a2=1

Exercise 8.7.47

36y2=1

Answer

y=6,6

Exercise 8.7.48

2x2+3x=2

Exercise 8.7.49

2a25a=3

Answer

a=13,2

Exercise 8.7.50

2x2+7x=6

Exercise 8.7.51

4a2+9a=9

Answer

a=13,43

Exercise 8.7.52

2x=3x+2+1

Exercise 8.7.53

1x=2x+432

Answer

x=43,2

Exercise 8.7.54

4m5m3=7

Exercise 8.7.55

6a+12a2=5

Answer

a=45,1

For the following problems, solve each literal equation for the designated letter.

Exercise 8.7.56

V=GMmD for D

Exercise 8.7.57

PV=nrt for n.

Answer

n=PVrt

Exercise 8.7.58

E=mc2 for m

Exercise 8.7.59

P=2(1+w) for w.

Answer

W=P22

Exercise 8.7.60

A=12h(b+B) for B.

Exercise 8.7.61

A=P(1+rt) for r.

Answer

r=APPt

Exercise 8.7.62

z=xˆxs for ˆx

Exercise 8.7.63

F=S2xS2y for S2y

Answer

S2y=S2xF

Exercise 8.7.64

1R=1E+1F for F.

Exercise 8.7.65

K=12h(s1+s2) for s2.

Answer

S2=2KhS1 or 2KhS1h

Exercise 8.7.66

Q=2mns+t for s.

Exercise 8.7.67

V=16π(3a2+h2) for h2.

Answer

h2=6V3πa2π

Exercise 8.7.68

I=ER+r for R.

Exercises For Review

Exercise 8.7.69

Write (4x3y4)2 so that only positive exponents appear.

Answer

y816x6

Exercise 8.7.70

Factor x416

Exercise 8.7.71

Supply the missing word. An slope of a line is a measure of the _____ of the line.

Answer

steepness

Exercise 8.7.72

Find the product x23x+2x2x12x2+6x+9x2+x2x26x+8x2+x6

Exercise 8.7.73

Find the sum. 2xx+1+1x3

Answer

2x25x+1(x+1)(x3)


This page titled 8.7: Rational Equations is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Denny Burzynski & Wade Ellis, Jr. (OpenStax CNX) .

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