1. Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu
Chabot Mathematics
§6.6 Rational Equations
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer

2. 6.4 Review § Any QUESTIONS About Any QUESTIONS About HomeWork
MTH 55
Review §
Any QUESTIONS About
§6.4 → Complex Rational Expressions
Any QUESTIONS About HomeWork
§6.4 → HW-26

3. Solving Rational Equations
In previous sections, we learned how to simplify expressions. We now learn to solve a new type of equation. A rational equation is an equation that contains one or more rational expressions. Some examples:
We want determine the value(s) for x that make these Equations TRUE

4. To Solve a Rational Equation
List any restrictions that exist. Numbers that make a denominator equal 0 canNOT possibly be solutions.
CLEAR the equation of FRACTIONS by multiplying both sides by the LCM of ALL the denominators present
Solve the resulting equation using the addition principle, the multiplication principle, and the Principle of Zero Products, as needed.
Check the possible solution(s) in the original equation.

5. 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
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.

6. Checking Answers
Since a variable expression could represent 0, multiplying both sides of an equation by a variable expression does NOT always produce an Equivalent Equation
COULD be Multiplying by Zero and Not Know it
Thus checking each solution in the original equation is essential.

7. Example  Solve
SOLUTION - Note that x canNOT equal 0. The Denominator LCM is 15x.

8.  Example  Solve CHECK tentative Solution, x = 5
The Solution x = 5 CHECKS 

9. Example  Solve
SOLUTION - Note that x canNOT equal 0. The Denom LCM is x
Thus by Zero Products:
x = or
x = 4

10. Example  Solve CHK: For x = 3 For x = 4
Both of these check, so there are two solutions; 3 and 4

11. Example  Solve
SOLUTION  Note that y canNOT equal 3 or −3. We multiply both sides of the equation by the Denom LCM.

12. Example  Solve
SOLUTION - Note that x canNOT equal 1 or −1. Multiply both sides of the eqn by the LCM
Because of the restriction above, 1 must be rejected as a solution. This equation has NO solution.

13. 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, 4x:
Multiplying by the LCD to clear fractions

14. Example  Solve SOLN cont. Using the distributive law
Locating factors equal to 1
Removing factors equal to 1
Using the distributive law

15. Example  Solve
SOLN cont.
This should check since x  0.
CHECK 8

16. Rational Eqn CAUTION
When solving rational equations, be sure to list any restrictions as part of the first step.
Refer to the restriction(s) as you proceed

17. 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

18. Example  Solve SOLUTION: By LCD Multiplication
Remove factors Equal to One and solve the resulting Eqn
Keep in Mind any restrictions

19. Example  Solve SOLN cont.: Multiply and Collect Similar terms
A check will confirm that 22 is the solution

20. Example  Eqn with NO Soln
Solve 3
x – 1 = 2
x + 1 – 6
x2 – 1
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.
Multiply each side by the LCD, (x –1)(x + 1). = 3
x – 1 2
x + 1 – 6
x2 – 1
(x – 1)(x + 1) = 3
x – 1 2
x + 1 – 6
x2 – 1
(x – 1)(x + 1) = – 6
3(x + 1)
2(x – 1)
Distributive property
Multiply. = – 6
3x + 3
2x + 2
Distributive property = 6
x + 5
Combine terms. = 1 x
Subtract 5.

21. Example  Eqn with NO Soln
Solve 3
x – 1 = 2
x + 1 – 6
x2 – 1
Since 1 is not in the domain, it cannot be a solution of the equation.
Substituting 1 in the original equation shows why. = 3
x – 1 2
x + 1 – 6
x2 – 1
Check: = 3
1 – 1 2
1 + 1 – 6
12 – 1 = 3 2 – 6
Since division by 0 is undefined, the given equation has no solution,
and the solution set is ∅.

22. Example  Fcn to Eqn Given Function: Find all values of a for which
SOLUTION  On Board
By Function Notation:
Thus Need to find all values of a for which f(a) = 4

23. Example  Fcn to Eqn Solve for a:
First note that a  0. To solve for a, multiply both sides of the equation by the LCD, a:
Multiplying both sides by a. Parentheses are important.
Using the distributive law

24. Example  Fcn to Eqn CarryOut Solution CHECK
Simplifying
Getting 0 on one side
Factoring
Using the principle of zero products
CHECK
STATE: The solutions are 5 and −1. For a = 5 or a = −1, we have f(a) = 4.

25. Rational Equations and Graphs
One way to visualize the solution to the last example is to make a graph. This can be done by graphing; e.g., Given
Find x such that f(x) = 4

26. Rational Equations and Graphs
Graph the function, and on the same grid graph y = g(x) = 4 4 -1 5
We then inspect the graph for any x-values that are paired with 4. It appears from the graph that f(x) = 4 when x = 5 or x = −1.

27. Rational Equations and Graphs
Graphing gives approximate solutions
Although making a graph is not the fastest or most precise method of solving a rational equation, it provides visualization and is useful when problems are too difficult to solve algebraically

28. WhiteBoard Work Problems From §6.6 Exercise Set Rational Expressions
34, 38, 62
Rational Expressions

29. Remember: can NOT Divide by ZERO
All Done for Today
Remember: can NOT Divide by ZERO

30. Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu
Chabot Mathematics
Appendix
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer –

31. Graph y = |x|
Make T-table