Solving Rational Equations: A Step-by-Step Guide, Slides of Algebra

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§6.6 Rational

Equations

Review §

 Any QUESTIONS About

• §6.4 → Complex Rational Expressions

 Any QUESTIONS About HomeWork

• §6.4 → HW-

MTH 55

To Solve a Rational Equation

1. List any restrictions that exist.

Numbers that make a denominator equal 0

canNOT possibly be solutions.

2. CLEAR the equation of FRACTIONS by

multiplying both sides by the LCM of ALL

the denominators present

3. Solve the resulting equation using the addition

principle, the multiplication principle, and the

Principle of Zero Products, as needed.

4. Check the possible solution(s) in the original

equation.

Example  Solve

 SOLUTION - Because no variable appears in the

denominator , no restrictions exist. The LCM of 5,

2, and 4 is 20, so we multiply both sides by 20

1

5 2 4

x x = −

20 20

1

5 2 4

^ x^   x  ⋅ (^)   = ⋅ (^)  −     

20 20 4

0

1

2

2 5

x x ⋅ = ⋅ − ⋅

4 x = 10 x − 5

− 6 x = − 5

5

6

x =

Using the multiplication principle to multiply both sides by the LCM. Parentheses are important!

Using the distributive law. Be sure to multiply EACH term by the LCM

Simplifying and solving for x. If fractions remain, we have either made a mistake or have not used the LCM of ALL the denominators.

Example  Solve

 SOLUTION - Note that x canNOT

equal 0. The Denominator LCM is 15 x.

1 1 4

3 x x 15

• =

x x

x x

 +^  =^ ⋅

5 + 15 = 4 x

20 = 4 x

5 = x

5

15 x

3 x

⋅ + 15 x

x

x ⋅

Example  Solve

 CHECK

tentative

Solution,

x = 5

1 1 4

3 x x 15

• =

3 x x 15

 The Solution

x = 5 CHECKS (^) 

Example  Solve

 CHK: For x = 3 For x = 4

12 x 7 x

• =

x 7

x

x 7

x

 Both of these check, so there are

two solutions; 3 and 4

Example  Solve

 SOLUTION  Note that y canNOT

equal 3 or −3. We multiply both sides of

the equation by the Denom LCM.

5 3 2

( 3)( 3

( 3)( 3) ( 3 ( 3 ) 3 3

y ) ) y y y y

y y y

      =^  − 

• −  −^   +^ −



−

( y + 3 )( y − 3 )

( y + 3 )( y − 3 )

( y 3

• )( 3)

3

y

y

−

( y + 3)( y − 3 −

)

y − 3

5 = 3( y − 3) − 2( y +3)

5 = 3 y − 9 − 2 y − 6

5 = y − 15

20 = y

2

5 3 2

y 9 y 3 y 3

= − − + −

Example  Solve

 SOLUTION: Because the left side of

this equation is undefined when x is 0,

we state at the outset that x ≠ 0.

 Next, we multiply both sides of the

equation by the LCD, 4 x :

2 7 5

4

x x

x x

− +

• =

Multiplying by the LCD to clear fractions

2 7 4 4 8

5

4

x x

x x

x x

 −^ +   +^  =  

⋅

Example  Solve

 SOLN cont.

2 7 5

4

x x

x x

− +

• =

x x

x x

x

x x

x x

x

x x

x x

Using the distributive law

Locating factors equal to 1

Removing factors equal to 1

Using the distributive law

(2 x − 7) + 4( x + 5) = 32 x

2 x − 7 + 4 x + 20 = 32 x

Rational Eqn CAUTION

• When solving rational equations, be sure

to list any Division-by-Zero restrictions as

part of the first step.

• Refer to the

restriction(s) as

you proceed

Example  Solve

 SOLUTION: To find all restrictions and

to assist in finding the LCD, we factor:

 Note that to prevent division by zero

x ≠ 3 and x ≠ −3.

 Next multiply by the LCD, ( x + 3)( x – 3),

and then use the distributive law

2

8 6 2 . x 3 x 3 x 9

− =

• − −

8 6 2 . x 3 x 3 ( x 3)( x 3)

− =

• − + −

Example  Solve

 SOLN cont.:

Multiply and

Collect

Similar terms

 A check will confirm that 22 is the solution

2

8 6 2 . x 3 x 3 x 9

− =

• − −

8( x − 3) − 6( x + 3) = 2

8 x − 24 − 6 x − 18 = 2

2 x − 42 = 2

2 x = 44

x = 22

Example  Eqn with NO Soln

To avoid division by zero, exclude from the expression domain 1 and –1, since these values make one or more of the denominators in the equation equal 0.

Distributive property

Solve 3. x – 1

=

2 x + 1

6 x^2 – 1

=

3 x – 1

2 x + 1

• 6 x^2 – 1

( x – 1)( x + 1) ( x – 1)( x + 1)

=

3 x – 1

2 x + 1

• 6 x^2 – 1

( x – 1)( x + 1) ( x^ – 1)( x^ + 1) ( x – 1)( x + 1)

3( x + 1) –^ 2( x – 1) = 6

3 x + 3 –^2 x + 2 = 6

x + 5 = 6

x = 1

Multiply each side by the LCD, ( x –1)( x + 1).

Multiply.

Distributive property

Combine terms.

Subtract 5.