1. Solving Rational Equations

2. Solving Equations Equation Expression Equation has an = sign
Expression does not

3. Solving Equations
Clear the Fractions: multiply both sides of the equation by LCD
Distribute
Solve
Check for EXTRANEOUS solutions. (ie. Is your answer an excluded value?)
Why does this work for equations, and not simplifying expressions?

4. Solving Equations Example Solve the following rational equation. true
LCD:
Check
true

5. Solving Equations
Example
Continued.

6. Solving Equations Example Continued
Notice 3 is an excluded value, so it’s an Extraneous Solution.
Check.
No Solution or

7. Problem Solving with Rational Equations

8. Solving a Work Problem Example 1
Joe can clean the house in 4 hours. Carl can clean the same house in 2 hours. How long will it take if they work together?
1.) Understand
By using the times for each to complete the job alone, we can figure out their work “rates per hour”
Joe /4
Carl /2
Time in hrs
Rate per hour
Continued

9. Solving a Work Problem Example continued Joe Carl Rate/hour alone time
together
In hours
Rate/hour
alone
time
together
in hours
1 Job
Continued

10. Solving a Work Problem Example 2
Joe can roof a house in 20 hours. If Carl helps Joe, it will take 11 hours to do the same job. How long will it take Carl to work alone?
1.) Understand
By using the times for each roofer to complete the job alone, we can figure out their work “rates per hour”
Joe /20
Carl x /x
Together Job
Time in hrs
Rate per hour
Continued

11. Solving a Work Problem Example continued Joe Carl Rate/hour alone time
together
In hours
Rate/hour
alone
time
together
in hours
1 Job
Continued

12. Solving a Rate Problem Example
The speed of Lazy River’s current is 5 mph. A boat travels 20 miles downstream in the same time as traveling 10 miles upstream. Find the speed of the boat in still water.
1.) Understand
Read and reread the problem. By using the formula d=rt, we can rewrite the formula to find that t = d/r.
rate of the boat downstream is the rate in still water + the water current and the rate of the boat upstream would be the rate in still water – the water current.
Down r /(r + 5)
Up r – /(r – 5)
Distance rate time = d/r
Continued

13. Solving a Rate Problem Example continued 2.) Translate
Since the problem states that the time to travel downstairs was the same as the time to travel upstairs, we get the equation
Continued

14. Solving a Rate Problem
Example continued
3.) Solve
Continued

15. Solving a Rate Problem Example continued 4.) Interpret
Check: We substitute the value we found from the proportion calculation back into the problem.
true
State: The speed of the boat in still water is 15 mph.

16. Simplifying Complex Fractions

17. Complex Rational Fractions
Complex rational expressions (complex fraction) are rational expressions whose numerator, denominator, or both contain one or more rational expressions.
There are two methods that can be used when simplifying complex fractions.

18. Simplifying Complex Fractions
Simplifying a Complex Fraction (Method 1)
Simplify the numerator and denominator of the complex fraction so that each is a single fraction.
Multiply the numerator of the complex fraction by the reciprocal of the denominator of the complex fraction.
Simplify, if possible.

19. Simplifying Complex Fractions
Example

20. Simplifying Complex Fractions
Method 2 for simplifying a complex fraction
Find the LCD of all the fractions in both the numerator and the denominator.
Multiply both the numerator and the denominator by the LCD.
Simplify, if possible.

21. Simplifying Complex Fractions
Example