Chapter 7 Rational Functions - College of the Redwoods

Section 7.7 Solving Rational Equations 717
y
10
y
10
y = 2 A B
x
10
x
10
(a)
Figure 6.
(b)
Solving f(x) = 2 graphically.
Next, let’s address the task required in part (c). We have very reasonable estimates
of the solutions of f(x) = 2 based on the data presented in Figure 6(b). Let’s use the
graphing calculator to improve upon these estimates.
First, load the equation Y2=2 into the Y= menu, as shown in Figure 7(a). We need
to find where the graph of Y1 intersects the graph of Y2, so we press 2nd CALC and
select 5:intersect from the menu. In the usual manner, select “First curve,” “Second
curve,” and move the cursor close to the point you wish to estimate. This is your
“Guess.” Perform similar tasks for the second point of intersection.
Our results are shown in Figures 7(b) and Figures 7(c). The estimate in Figure 7(b)
has x ≈ 0.43844719, while that in Figure 7(c) has x ≈ 4.5615528. Note that these
are more accurate than the approximations of x ≈ 0.3 and x ≈ 4.6 captured from our
hand drawn image in Figure 6(b).
(a) (b) (c)
Figure 7.
Solving f(x) = 2 graphically.
Finally, let’s address the request for an algebraic solution of f(x) = 2 in part (d).
First, replace f(x) with 1/x + 1/(x − 4) to obtain
f(x) = 2
1
x + 1
x − 4 = 2.
Multiply both sides of this equation by the common denominator x(x − 4).
Version: Fall 2007