Practice Makes Perfect Algebra - Carolyn Wheater [28]
• Factor each of the denominators.
• Determine the LCD of all the algebraic fractions in the equation. The LCD of x(x + 2),(x + 1)(x + 2), and x(x + 1) is x(x + 2)(x + 1).
• Multiply both sides of the equation by the LCD, distributing as necessary.
• Cancel as you multiply, and all denominators should disappear.
• Solve the resulting equation, and check your solution.
(x + 1)(x + 1) + x·3 =(x + 2)(x + 4)
x2 + 2x + 1 + 3x = x2 + 6x + 8
5x + 1 = 6x + 8
−7 = x
Check:
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EXERCISE 11.7
Solve each rational equation by multiplying through by the LCD.
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Graphing rational functions
The graphs of rational functions are often discontinuous—that is, the graph is in two or more pieces—because the function is undefined for any value that makes the denominator equal to 0. Simple rational functions have a characteristic two-wing shape called a hyperbola as shown in Figure 11.1, but more complicated rational functions have various graphs. You’ll want to make a table of values; there are a few tips that can help you choose useful x-values.
Figure 11.1 The graph of a simple rational function.
Vertical asymptotes
An asymptote is a line that is not part of the graph, but one that the graph approaches closely. When the graph gets close to the vertical asymptote, it curves either upward or downward very steeply so that it looks almost vertical itself. Remember that the graph can get very close to the asymptote but can’t touch it.
Vertical asymptotes occur at many of the values of x for which the function is undefined, so before you begin to build a table of values, find the value(s) of x that would make the denominator equal to 0. These will be discontinuities, or breaks, in the graph, and you can expect that as you get near these x-values, the y-values will become very large(positive) or very small(negative). Choose x-values on both sides of the vertical asymptotes. The function has a vertical asymptote of x = −1 and has a vertical asymptote of x = 2 as shown in Figure 11.2.
Figure 11.2 Graphs of and .
Horizontal asymptotes
Horizontal asymptotes, like vertical asymptotes, are not actually part of the graph, but they are lines that the left and right ends of the graph approach closely. The graphs of simple rationals often flatten out on the ends, so these asymptotes are often horizontal lines.
If the degree of the numerator is less than the degree of the denominator, as in , the horizontal asymptote will be y = 0. If the degree of the numerator and denominator are the same, as in , divide the lead coefficients to find the horizontal asymptote. The horizontal asymptote for is y = 2.
Drawing the vertical and horizontal asymptotes as dotted lines before you begin graphing will help you locate the graph.
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EXERCISE 11.8
Graph each rational function. Sketch the vertical and horizontal asymptotes first.
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Work problems
Problems that deal with how long it takes two people(or two machines) to do a job while working together can be organized much the same way as the mixture problems you saw earlier, if you know one little trick. As soon as you are told how long it takes someone to do the job, express the part of the job they can do in one unit of time—1 min, 1 h, 1 day, whatever.
Suppose George can do the job in 2 h and Harry can do it in 3 h. How long will it take them to do it working together? You want to set up a table as you did with the other problems, but it will look like this.
The part of the job done in 1 h, times the number of hours, will always equal 1. Your equation comes not from adding the hours, but from adding the part of the job done in an hour.
It will take hours, or 1 h and 12 min, for them to do the job together.
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EXERCISE 11.9
Solve each problem and check your answer.
1. Marilina can mow the lawn in 3 h, and Jeanne can mow the same lawn in 5 h. How long will it take to mow the lawn if they work together?(You can