Showing posts with label rational. Show all posts
Showing posts with label rational. Show all posts

Wednesday, November 7, 2012

Chapter 7 Sample Test Questions

Click here for a worksheet containing 20 sample test questions with answers.

Word Problems
Example: The sum of the reciprocals of two consecutive odd integers is 4/3. Set up an algebraic equation and use it to find the two integers.
Example: Norm can paint the office by himself in 5 hours.  Cliff could do the same job in 7 hours. If they work together, how long will it take them to paint the office?
Example: An executive went on an 8-hour business trip that required a 165-mile bus ride to the airport then another 2,750 miles by airplane.  If the airplane speed was 10 times that of the bus, what is was the average speed of the airplane?

Tuesday, November 6, 2012

Solving Rational Equations

Rational equations are simply equations with rational expressions in them. Use the technique outlined earlier to clear the fractions of a rational equation.  After clearing the fractions, we will be left with either a linear or a quadratic equation that can be solved as usual.

   Step 1: Factor the denominators.
   Step 2: Identify the restrictions.
   Step 3: Multiply both sides of the equation by the LCD.
   Step 4: Solve as usual.
   Step 5: Check answers against the set of restrictions.
This process sometimes produces answers that do not solve the original equation (extraneous solutions), so it is extremely important to check your answers.
Tip: It suffices to check that the answers are not restrictions to the domain of the original equation.

Determining the LCD is often the most difficult part of the process. Use one of each factor found in all denominators.
Because of the distributive property, multiplying both sides of an equation by the LCD is equivalent to multiplying each term by that LCD as illustrated in the following examples.
Some literal equations, often referred to as formulas, are also rational equations. Use the techniques of this section and clear the fractions before solving for the particular variable.

Solve for the specified variable.
The reciprocal of a number is the number we obtain by dividing 1 by that number.

Word Problem: The reciprocal of the larger of two consecutive positive odd integers is subtracted from twice the reciprocal of the smaller and the result is 9/35. Find the two integers.
Video Examples on YouTube:

Complex Rational Expressions

It turns out that we have all the tools necessary to simplify complex algebraic fractions. The numerator and denominator of these rational expressions contain fractions and look very intimidating.  We will outline two methods for simplifying them.

Method 1: Obtain a common denominator for the numerator and denominator, multiply by the reciprocal of the denominator, then factor and cancel if possible.
Method 2: Multiply the numerator and denominator of the complex fraction by the LCD of all the simple fractions then factor and cancel if possible.

To illustrate what happened after we multiplied by the LCD we could add an extra step.

For the following solved problems, both methods are used. Choose whichever method feels most comfortable for you.

Simplify using method 1.                                 Simplify using method 2.
Video Examples on YouTube: