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This lesson focuses on solving equations that contain one or more rational expressions. Since methods for solving this type of equations has already been covered, this lesson will focus on inequalities that contain rational expressions.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
Challenge
Dominika's Winning Percentage in Basketball
Heichi and Dominika like to play basketball. About two months ago, they decided to keep track of how many games they each win. Until now, Dominika has won out of the games against Heichi.
Error loading image.
a How many games would Dominika have to win in a row to have a winning record?
b How many games would Dominika have to win in a row to have a winning record?
c Is Dominika able to reach a winning record? Explain why or why not.
d Suppose that after reaching a winning record of Dominika had a losing streak. How many games in a row would Dominika have to lose to drop down to a winning record below again?
Discussion
Rational Equations and Finding Their Solutions
A rational equation is an equation that contains at least one rational expression.
If a rational equation is expressed as a proportion, it can be solved by using the Cross Products Property. Otherwise, to eliminate the fractions, each side of the equation is multiplied by the least common denominators of the expressions.
When a rational equation has been solved using algebraic methods, it is necessary to ensure that each solution satisfies the original equation because extraneous solutions may appear.
| Rational Equation | Method |
|---|---|
| Cross Products Property | |
| LCD |
Method
Solving a Rational Equation by Using the Cross Products Property
Rational equations can be solved by using different methods. The main idea is to move the variables out of the denominators. Furthermore, since the domain of rational functions is restricted, some solutions might be extraneous. Therefore, it is necessary to verify the solutions in the original equation. Consider this rational equation as an example.
The equation is basically a proportion. Therefore, it can be solved by using the Cross Products Property.
1
Use the Cross Products Property
expand_more To get the variable terms out of the denominators, the Cross Product Property will be used.
As shown, the property eliminates the rational expressions in the equation. Now the equation can be solved for
2
Solve the Resulting Equation
expand_more To solve the resulting equation, use the Distributive Property and combine like terms.
The value that satisfies the equation is Next, check if it also satisfies the original equation.
3
Check for Extraneous Solutions
expand_more The solution found in the previous step might be extraneous. It must be verified in the original equation.
Substiting in the original equation produced a true statement. Therefore, it is a solution to the original equation.
▼
Evaluate right-hand side
\(1 = 1 \ \ {\color{#009600}{\bm{\Large{\checkmark }}}}\)
Example
Creating a Acid Solution
In her chemistry lab, Dominika adds some acid solution to milliliters of a solution with acid.
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Hint
The percentage of acid in the final solution must equal the total amount of acid divided by the total amount of solution.
Solution
Let be the amount of acid solution to be added to milliliters of a acid solution. The amount of each solution in terms of can be organized in a table.
| Original | Added | New | |
|---|---|---|---|
| Amount of Acid | |||
| Total Solution |
\(0.45 = 0.45 \ \ {\color{#009600}{\bm{\Large{\checkmark }}}}\)
Discussion
Solving a Rational Equation by Using the Least Common Denominator
To solve a rational equation, the first step is to eliminate all the fractions in the equation. This can be done by multiplying each side of the equation by the least common denominator of the rational expressions. Any time an equation is multiplied by an algebraic expression, an extraneous solution might be introduced, so it will also be necessary to verify the solutions.
Consider solving the this equation as an example.
1
Find the Least Common Denominator
expand_more To find the least common denominator (LCD) of the rational expressions in the equation, their factored form must be found.
The denominator of the rational expression on the left-hand side cannot be factored further. Factor the denominator of the other expression.
Since the LCD is the product of the factors with the highest power appearing in any denominator, the LCD is
| Rational Expression | LCD |
|---|---|
2
Multiply the Equation by the Least Common Denominator
expand_more Multiply each side of the equation by the LCD found in the previous step. This will reduce each term and get rid of the denominators.
▼
Simplify
3
Solve the Resulting Equation
expand_more Now the resulting equation can be solved by using the Distributive Property and inverse operations.
4
Check for Extraneous Solutions
expand_more Finally, check if the solution found in Step is an extraneous solution or not. To do so, substitute it in the original equation and simplify.
The value made a true statement. Therefore, it is a solution to the equation.
▼
Evaluate right-hand side
\(4 = 4 \ {\color{#009600}{\bm{\Large{\checkmark }}}}\)
Example
Canoe Trip
Dominika and Heichi are taking a canoe trip. They are going up the river for kilometer and then returning to their starting point. The river current flows at kilometers per hour. The total trip time will be hour and minutes.
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Hint
Let represent the speed, in kilometers per hour, that the canoe would travel with no current. Write two rational expressions for the time it takes to go and return in terms of
Solution
The two students are on the river for hour and minutes, or hours. This round trip time represents the time it takes to go upriver and return. Recall the formula that relates distance to time, This formula can be maniuplated to represent distance in terms of time.
Let represent the speed, in kilometers per hour, that the canoe would travel with no current. When Dominika and Heichi are traveling with the current, their speed is and when they travel against the current, their speed is The trip takes hours for the outbound portion and hours for the return portion.
The sum of the portions is equal to the total round trip time.
The least common denominator of the rational expressions is To eliminate the rational denominators, multiply each side of the equation by the LCD.
To get rid of decimals in the equation, both sides of the equation can be multiplied by The equation can then be solved by using the Quadratic Formula.
The solutions for this equation are Separate them into the positive and negative cases and round the answers to two decimal places.
The second solution does not make sense in the given context as speed cannot be negative. The first solution must be verified in the original equation to determine if it is extraneous or not. If it is a valid solution, substituting should equal about
Substituting the solution into the original equation resulted in an identity, so this is a solution to the original equation. Therefore, the speed at which the students can row their canoe is about kilometers per hour.
| Going | Returning | |
|---|---|---|
| Distance (km) | ||
| Speed (km/h) | ||
| Time (h) |
▼
Simplify left-hand side
Distr
Distribute
CancelCommonFac
Cancel out common factors
SimpQuot
Simplify quotient
AddTerms
Add terms
▼
Simplify right-hand side
RearrangeEqn
Rearrange equation
▼
Solve using the quadratic formula
UseQuadForm
Use the Quadratic Formula:
CalcPowProd
Calculate power and product
AddTerms
Add terms
SplitIntoFactors
Split into factors
CalcRoot
Calculate root
FactorOut
Factor out
SimpQuot
Simplify quotient
RoundInt
Round to nearest integer
\(1.5 = 1.5 \ {\color{#009600}{\bm{\Large{\checkmark }}}}\)
Example
Identifying an Extraneous Solution
Heichi is trying to solve the following rational equation.
However, he cannot be sure about his solution, so he shares his thoughts with Dominika.
Multiplying both sides by is legitimate unless or When or this is equivalent to multiplying both sides by Therefore, the operation only produces an equivalent equation if those two values are excluded.
After the equation was solved for it was found that is a solution. However, since is excluded, it is an extraneous solution.
|
If I clear the denominators I find that the only solution is but when I substitute into the equation, it does not make any sense. |
Answer
Yes, see solution.
Hint
Solve the given rational equation. Is the solution in the domain of the rational expressions on both sides?
Solution
To check if Heichi's claim is correct or not, the given equation will be solved by using the least common denominator. Notice that so is the common denominator for the three rational expressions. Multiply the equation by this common denominator and solve for
Heichi is correct that when the denominators are eliminated, the solution to the resulting equation is However, the rest of his statement is also correct because the denominators of the expressions on the left hand side are when
Therefore, the value Heichi found is not in the domain of the rational expressions on the left-hand side, which makes it an extraneous solution.
▼
Solve for
Distr
Distribute
CancelCommonFac
Cancel out common factors
SimpQuot
Simplify quotient
Distr
Distribute
RearrangeEqn
Rearrange equation
Extra
Why is an extraneous solution?An alternative explanation for why are extraneous solutions can be explained as follows. In the example, both sides of the equation are multiplied by to create an equivalent equation.
Example
Finding the Time Spent Painting the Canoe
Dominika and Heichi want to paint their canoe before their next trip up the river.
a Dominika thinks that if she works alone, it would take her times as long as it would take Heichi to paint the entire canoe by himself. However, by working together, they can complete the job in hours. How long would it take Heichi to complete the job?
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b Heichi thinks that if he works alone, it would take him hours less than it would take Dominika to paint the entire canoe by herself. If they worked together, they could complete the job in hours. How long would it take Dominika to paint the canoe if Heichi were not there to help?
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Hint
a Let be the time it takes Heichi to complete the work. The amount of work done is the product of the rate of work and the time spent working
b The amount of work done is the product of the rate of work and the time spent working
Solution
a Let be the time it takes Heichi to complete the work alone. Then, the time it would take Dominika to complete the work by herself would be The work in this case is painting canoe, or Now an expression to represent each person’s rate can be written using the formula
▼
Solve for
▼
Evaluate left-hand side
\(1 = 1 \ {\color{#009600}{\bm{\Large{\checkmark }}}}\)
b Let be the time it takes Dominika to complete the work. Then, the time it would Heichi to complete the same amount of work would be The work in this case is painting canoe, or Now an expression to represent each person’s rate can be written using the formula
▼
Solve using the quadratic formula
UseQuadForm
Use the Quadratic Formula:
CalcPowProd
Calculate power and product
SubTerm
Subtract term
CalcRoot
Calculate root
▼
Evaluate left-hand side
\(1 = 1 \ {\color{#009600}{\bm{\Large{\checkmark }}}}\)
Discussion
Rational Inequalities and Finding Their Solution Sets
A rational inequality is an inequality that contains a rational expression.
The general form of a rational inequality has a rational expression on the left-hand side of the inequality and on the right-hand side of the inequality.
The steps followed to solve a rational inequality are quite similar to those followed to solve linear inequalities.
Method
Solving a Rational Inequality
To solve a rational inequality, the excluded values are first identified. The related rational equation is solved, then the excluded values and solutions of the equation are used to divide the number line into intervals. Finally, the solution set is determined by testing a value in each interval. Consider the following rational inequality.
To solve this inequality, these steps can be followed.
1
Identify the Excluded Values
expand_more Take a look at the denominators of the rational expressions in the inequality.
The excluded values are the ones that make the denominator
| Denominator | Factored Form | Excluded Values |
|---|---|---|
| and | ||
The excluded values for this inequality are and
2
Solve the Related Rational Equation
expand_more The related rational equation can be written by replacing the inequality sign with an equals sign.
To solve this rational equation, all of the denominators must be eliminated. Notice that is the least common denominator.
Now that an equation without any denominators is obtained, it can be solved for
This is another value of interest.
▼
Simplify
3
Use the Values Found to Divide the Number Line Into Intervals
expand_more Next, the number line will be divided into intervals. The intervals are determined by the excluded values and and the solution to the related equation
4
Test a Value in Each Interval
expand_more Finally, a value in each interval will be tested to determine which intervals contain values that satisfy the inequality. Start with the interval The arbitrary point can be used. If substituting this point into the original inequality produces a true statement, then the numbers in this interval are solutions. If not, they are not solutions.
Solving for resulted in a false statement. Therefore, the numbers less than are not solutions to the inequality. The same process can be applied to the other intervals to determine whether or not they are solutions to the inequality.
The solutions to the given inequality are all the real numbers between and or all the numbers greater than
Notice that the given inequality is a non-strict inequality. Consequently, some of the endpoints might satisfy the inequality. The solution of the related equation, it also results in a true statement. Therefore, the square bracket is used for that endpoint to indicate that it is also a solution.
This solution set can also be shown on the number line. 
▼
Evaluate left-hand side
CalcPow
Calculate power
AddSubTerms
Add and subtract terms
MoveNegDenomToNum
Put minus sign in numerator
AddFrac
Add fractions
\(\dfrac{\text{-} 29}{7} \ngeq \dfrac{3}{7} \ {\color{#FF0000}{\bm{\Large{\times }}}}\)
| Test Value | ||||
|---|---|---|---|---|
| Statement |
Example
The Average Cost of a Basketball
The cost of producing basketballs is represented by
Since substituting into the inequality resulted in a true statement, it can be concluded that the numbers less than are solutions to the inequality. Use the same process for the other intervals to find any other solutions.
The solutions to the inequality are all the numbers below or above
However, cannot be negative because it represents the number of basketballs produced. Therefore, that part of the solution can be ignored. Additionally, there is no need to check for endpoints as the inequality is strict. As a result, the average cost will be less than as long as more than basketballs are produced.
a Find the average cost function,
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b How many basketballs should be produced so that the average cost is less than Write the answer as an inequality.
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Hint
a Divide the cost function by
b Start by writing a rational inequality. Then, solve the rational inequality. Do negative values make sense in this context?
Solution
a The average cost function is found by dividing the cost function by the total number of items produced. The average cost function of producing basketballs is then
b The values of that make the average cost less than need to be found. This can be represented algebraically by an inequality.
- State the excluded values.
- Solve the related rational equation.
- Use the values determined in the previous steps to divide a number line into intervals.
- Test a value in each interval to determine which intervals contain values that satisfy the inequality.
Excluded Values
Consider the inequality.Solve the Related Equation
The related equation is obtained by replacing the inequality sign with an equals sign.Divide the Number Line Into Intervals
Next, the number line will be divided into intervals using the excluded value and the solution to the related equation
Test Values
Select an arbitrary value from each interval to see if it produces a true statement. Starting with the interval the point will be tested. If substituting this point into the inequality produces a true statement, then the numbers in this interval are solutions. If not, they are not solutions.▼
Evaluate left-hand side
\(\text{-} 1995 < 30 \ {\color{#009600}{\bm{\Large{\checkmark }}}}\)
| Choose a Test Value | |||
|---|---|---|---|
| Statement |
Closure
Calculating Dominika’s Winning Percentages for Different Situations
When solving rational equations, the goal is to eliminate the rational denominators. However, the mechanics of solving rational inequalities are quite different. With all the methods discussed in this lesson, the challenge presented at the beginning can be reconsidered now. Dominika and Heichi keep track of how many games each of them win.
Error loading image.
Given that Dominika has won out of the games against Heichi, the following situation will be analyzed.
a How many games would Dominika have to win in a row to have a winning record?
b How many games would Dominika have to win in a row to have a winning record?
c Is Dominika able to reach a winning record? Explain why or why not.
d Suppose that after reaching a winning record of Dominika had a losing streak. How many games in a row would Dominika have to lose to drop down to a winning record below again?
Answer
a games
b games
c No, see solution.
d More than games
Hint
a Let be the number of games that Dominika wins. Write a proportion using
b Repeat the same procedure as in the previous section.
c Is there a scenario where Dominika wins all the games?
d Consider the answer found in Part B. Write a rational inequality for the number of games that Dominika loses without winning any.
Solution
a The ratio of Dominika's wins to the total number of games is currently
\(0.75 = 0.75 \ {\color{#009600}{\bm{\Large{\checkmark }}}}\)
b The number of games that Dominika has to win can be found by following the same process as in the previous part. In this case, Dominika wants to win of the games. Therefore, the ratio of to should be equal to
\(0.90 = 0.90 \ {\color{#009600}{\bm{\Large{\checkmark }}}}\)
c In order for Dominika to have won of the games, she must have won all of them. However, this is not the case because she has already lost out of the games played.
| Dominika | |
|---|---|
| Win | Loss |
d Earlier it was determined that Dominika has to win consecutive games in order to bring her winning record up to In other words, Dominika will have to win out of games.
Finally, a value from each interval is chosen and tested. If substituting this point into the inequality produces a true statement, then the numbers in this interval are solutions. If not, these numbers are not solutions.
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