Chapter 7 Rational Functions - College of the Redwoods

Section 7.7 Solving Rational Equations 721
so
x = −4 or x = 2.
The fact that we have found two answers using an analytical method is troubling.
After all, the graph in Figure 10(b) indicates only one solution, namely x = −4. It is
comforting that one of our analytical solutions is also x = −4, but it is still disconcerting
that our analytical approach reveals a second “answer,” namely x = 2.
However, notice that we haven’t paid any attention to the restrictions caused by
denominators up to this point. Indeed, careful consideration of equation (11) reveals
factors of x+2 and x−2 in the denominators. Hence, x = −2 and x = 2 are restrictions.
Note that one of our answers, namely x = 2, is a restricted value. It will make some
of the denominators in equation (11) equal to zero, so it cannot be a solution. Thus,
the only viable solution is x = −4. One can certainly check this solution by hand, but
let’s use the graphing calculator to assist us in the check.
First, enter -4, press the STO◮ button, press X, then press ENTER to execute the
resulting command and store -4 in the variable X. The result is shown in Figure 11(a).
Next, we calculate the value of the left-hand side of equation (11) at this value
of X. Enter the left-hand side of equation (11) as 1/(X+2)-X/(2-X), then press the
ENTER key to execute the statement and produce the result shown in Figure 11(b).
Finally, enter the right-hand side of equation (11) as (X+6)/(xˆ2-4) and press
the ENTER key to execute the statement. The result is shown in Figure 11(c). Note
that both sides of the equation equal .1666666667 at X=-4. Thus, the solution x = −4
checks.
(a) (b) (c)
Figure 11. Using the graphing calculator to check the solution x = −4 of equation (11).
Version: Fall 2007