Rational Equations

I.  Return Midterm III 
II.  Rational Equations

Step by Step Process

Step 1:  Multiply Through by the Common Denominator

Step 2:  Bring everything over to the left hand side

Step 3:  Use the previously learned methods to solve the polynomial.

Step 4:  Check

Example:

Solve 1/(x + 1) - 1/(x - 1) = 2x/(x2 - 1) 

Step 1:  We multiply by x2 - 1

(x - 1) - (x + 1) = 2x

Step 2:  (x - 1) - (x + 1) - 2x = 0

-2x - 2 = 0

Step 3:  x = -1

Step 4:  We see that if x = -1, then the denominator of the original problem is zero, hence x = -1 is an extraneous solution.  We can conclude that there is no solution to this problem.

Exercises:  

A)  3/(x - 2) - 2/(2 - 3x) = 2/(3x2 - 8x + 4)

B)  (x + 1)/(x - 2) + (x - 1)/x + 2) = 3x/(x2 - 4) + 3/x

C)  5/(x + 2) + 2x/3 = (x - 6)/x

Application

You can canoe 8 miles down the Truckee River in twice the time it takes to canoe the same distance downstream.  In still water, you can canoe at 3 miles per hour.  How fast is the current?

III.  Rational Inequalities

Step by Step Method:

Step 1)  Bring everything to the left hand side of the inequality.

Step 2)  Put everything over a common denominator

Step 3)  Factor the top and the bottom and get the roots of each.

Step 4)  Place all roots on the number line to to cut the number line into several regions

Step 5)  Make a table with the regions and the factors from the numerator and denominator.

Step 6)  Put in test values to determine the positivity of each region.

Step 7)  Depending on whether the inequality is "<" or ">" choose the appropriate regions.

Step 8)  If the inequality is ">" or "<" include only those endpoints that come from the numerator.  If the inequality is "<" or ">" do not include any endpoints.

Example:

(x - 4)/(2x + 1) > 3

1)  (x - 4)/(2x + 1) - 3  > 0

2)   (x - 4)/(2x + 1) - 3(2x + 1)/(2x + 1)  > 0

 [(x - 4) - 3(2x + 1)]/(2x + 1)  > 0

[(x - 4) - 6x - 3]/(2x + 1)  > 0

(-5x - 7)/(2x + 1) > 0

3)  We have roots:  -7/5 and -1/2

4)  We see that the roots cut the number line into three regions

5 and 6)  We make the table  

-5x-7 2x+1 Total
I(-2) + - -
II(-1) - - +
III(0) - + -

7)  We choose region II.

8)  [-7/5,-1/2)

Exercise:

A)  x + (x - 3)/(2x + 5) < 3

B)  (5x + 3)/(x2 + 2x + 5) > 3x/(2x2 - 2)