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600 Chapter 8 Rational and Radical Functions
A rational equation is an equation that contains one or more rational expressions. The time t in hours that it takes to travel d miles can be determined by using the equation t = __ d r , where r is the average rate of speed. This equation is a rational equation.
To solve a rational equation, start by multiplying each term of the equation by the least common denominator (LCD) of all of the expressions in the equation. This step eliminates the denominators of the rational expressions and results in an equation you can solve by using algebra.
E X A M P L E 1 Solving Rational Equations
Solve the equation x + _^8 x = 6.
x ( x ) + _^8 x ( x ) = 6 ( x ) Multiply each term by the LCD, x. x^2 + 8 = 6 x Simplify. Note that x ≠ 0.
x^2 - 6 x + 8 = 0 Write in standard form.
( x - 2 )( x - 4 ) = 0 Factor.
x - 2 = 0 or x - 4 = 0 Apply the Zero Product Property.
x = 2 or x = 4 Solve for x.
Check −−−−−−
x + _^8 x = 6 −−−−−−
x + _^8 x = 6
2 + _^8 2
6 4 + _^8
Solve each equation.
1a. _^10 3
= _^4 x + 2 1b. _^6 x + _^5 4
= -_^7
1c. x = _^6 x - 1
An extraneous solution is a solution of an equation derived from an original equation that is not a solution of the original equation. When you solve a rational equation, it is possible to get extraneous solutions. These values should be eliminated from the solution set. Always check your solutions by substituting them into the original equation.
Solving Rational Equations
and Inequalities
Objective Solve rational equations and inequalities.
Vocabulary rational equation extraneous solution rational inequality
Who uses this? Kayakers can use rational equations to determine how fast a river is moving. (See Example 3.)
Factoring is not the only method of solving the quadratic equation that results in Example 1. You could also complete the square or use the Quadratic Formula.
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TEKS 2A.10.D Rational functions: determine the solutions of rational equations using graphs, tables, and algebraic methods.
Also 2A.2.A, 2A.10.A, 2A.10.B, 2A.10.C, 2A.10.E, 2A.10.F
8- 5 Solving Rational Equations and Inequalities 601
E X A M P L E 2 Extraneous Solutions
Solve each equation. A _^3 x x - 3
_^2 x^ +^3 x - 3
_^3 x x - 3
( x - 3 ) = 2 _ x^ +^3
x - 3
( x - 3 ) Multiply each term by the LCD, x - 3.
_^3 x x - 3
( x - 3 ) = _^2 x^ +^3
x - 3
( x - 3 ) Divide out common factors.
3 x = 2 x + 3 Simplify. Note that x ≠ 3.
x = 3 Solve for x. The solution x = 3 is extraneous because it makes the denominators of the original equation equal to 0. Therefore, the equation has no solution.
Check Substitute 3 for x in the original equation.
_^3 (^3 )
_^2 (^3 )^ +^3
_^9
_^9
✘ Division by 0 is undefined.
B _^2 x^ -^9 x **- 7
_^2 x^ -^9 x - 7
· 2 ( x - 7 )^ + _ x
· 2 ( x - 7 )^ = _^5
x - 7
· 2 ( x - 7 )
Multiply each term by the LCD, 2 ( x - 7 ). (^2) _ x - 9 x - 7
· 2 ( x - 7 )^ + _ x
· 2 ( x - 7 )^ = _^5
x - 7
· 2 ( x - 7 )^ Divide out
common factors.
2 ( 2 x - 9 )^ + x ( x - 7 )^ = 5 ( 2 )^ Simplify. Note that x ≠ 7.
4 x - 18 + x^2 - 7 x = 10 Use the Distributive Property. x^2 - 3 x - 28 = 0 Write in standard form.
( x - 7 )( x + 4 ) = 0 Factor.
x - 7 = 0 or x + 4 = 0
Use the Zero Product Property. x = 7 or x = - 4 Solve for x. The solution x = 7 is extraneous because it makes the denominators of the original equation equal to 0. The only solution is x = -4.
Check Write 2 _____ xx - - 79 + __ x 2 = ____ x -^5 7 as _____^2 x^ -^9 x - 7 +^
__ x 2 -^
____^5 x - 7 =^ 0. Graph the left side of
the equation as Y1 and identify the values
of x for which Y1 = 0.
The graph intersects the x -axis only when x = -4. Therefore, x = -4 is the only solution.
Solve each equation.
2a. _^16 x^2 - 16
= _^2
x - 4 2b. _^1 x - 1 = _ x x - 1
A rational expression is undefined for any value of a variable that makes a denominator in the expression equal to 0.
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8- 5 Solving Rational Equations and Inequalities 603
E X A M P L E 4 Work Application
Jason can clean a large tank at an aquarium in about 6 hours. When Jason and Lacy work together, they can clean the tank in about 3.5 hours. About how long would it take Lacy to clean the tank if she works by herself? Jason’s rate: _^1 6 of the tank per hour
Lacy’s rate: _^1 h
of the tank per hour, where h is the number of hours needed to clean the tank by herself Jason’s rate × hours worked + Lacy’s rate × hours worked = 1 complete job _ 1 6
(3.5) (^) + _^1 h
_^1
(3.5) ( 6 h ) (^) + _^1 h
(3.5) ( 6 h ) (^) = 1 ( 6 h ) (^) Multiply by the LCD, 6h.
3.5 h + 21 = 6 h Simplify. 21 = 2.5 h Solve for h. 8.4 = h It will take Lacy about 8.4 hours, or 8 hours 24 minutes, to clean the tank when working by herself.
4. Julien can mulch a garden in 20 minutes. Together, Julien and Remy can mulch the same garden in 11 minutes. How long will it take Remy to mulch the garden when working alone?
A rational inequality is an inequality that contains one or more rational expressions. One way to solve rational inequalities is by using graphs and tables.
E X A M P L E 5 Using Graphs and Tables to Solve Rational Equations and Inequalities
Solve _ x x - 4 ≤ 2 by using a graph and a table.
Use a graph. On a graphing
calculator, let Y1 = ____ x - x 4 and
Y2 = 2.
The graph of Y1 is at or below
the graph of Y2 when x < 4 or
when x ≥ 8.
Use a table. The table shows that
Y1 is undefined when x = 4 and
that Y1 ≤ Y2 when x < 4 or when
x ≥ 8.
The solution of the inequality is x < 4 or x ≥ 8.
Solve by using a graph and a table.
5a. _ x x - 3 ≥ 4 5b. _^8 x + 1
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x | x < 4 x ≥ 8
603
604 Chapter 8 Rational and Radical Functions
You can also solve rational inequalities algebraically. You start by multiplying each term by the least common denominator (LCD) of all the expressions in the inequality. However, you must consider two cases: the LCD is positive or the LCD is negative.
E X A M P L E 6 Solving Rational Inequalities Algebraically
Solve the inequality _^8 x + 5
≤ 4 algebraically.
Case 1 LCD is positive. Step 1 Solve for x. _^8 x + 5 ( x + 5 ) ≤ 4 ( x + 5 )
Multiply by the LCD.
8 ≤ 4 x + 20 Simplify. Note that x ≠ -5.
- 12 ≤ 4 x Solve for x.
- 3 ≤ x x ≥ - 3 Rewrite with the variable on the left.
Step 2 Consider the sign of the LCD. x + 5 > 0 LCD is positive. x > - 5 Solve for x.
For Case 1, the solution must satisfy x ≥ - 3 and x > - 5, which simplifies to x ≥ - 3.
Case 2 LCD is negative. Step 1 Solve for x. _^8 x + 5 ( x + 5 ) ≥ 4 ( x + 5 ) Multiply by the LCD. Reverse the inequality. 8 ≥ 4 x + 20 Simplify. Note that x ≠ -5.
- 12 ≥ 4 x Solve for x.
- 3 ≥ x x ≤ - 3 Rewrite with the variable on the left.
Step 2 Consider the sign of the LCD. x + 5 < 0 LCD is negative. x < - 5 Solve for x.
For Case 2, the solution must satisfy x ≤ - 3 and x < - 5, which simplifies to x < - 5.
The solution set of the original inequality is the union of the solutions to both Case 1 and Case 2. The solution to the inequality ____ x +^8 5 ≤ 4 is x < - 5 or x ≥ - 3, or
x ⎪ x < - 5 x ≥ - 3
Solve each inequality algebraically.
6a. _^6 x - 2
≥ - 4 6b. _^9 x + 3
THINK AND DISCUSS
- Explain why multiplying both sides of a rational equation by the LCD eliminates all of the denominators.
- Explain why rational equations may have extraneous solutions.
- Describe two methods for solving the inequality __^12 x > 3.
- GET ORGANIZED Copy and complete the graphic organizer. In each box, write the appropriate information related to rational equations.
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If you multiply or divide both sides of an inequality by a negative value, you must reverse the inequality symbol.
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