Chapter 7 Rational Functions - College of the Redwoods

Section 7.7 Solving Rational Equations 713
5
x + 6 x 2 = 5
3/2 + 6
(3/2) 2 = 5
3/2 + 6
9/4 .
In the final expression, multiply top and bottom of the first fraction by 2, top and
bottom of the second fraction by 4.
5
3/2 · 2
2 + 6
9/4 · 4
4 = 10
3 + 24
9
Make equivalent fractions with a common denominator of 9 and add.
10
3 · 3
3 + 24
9 = 30
9 + 24
9 = 54
9 = 6
Note that this result is identical to the left-hand side of the original equation (4).
Thus, x = 3/2 checks.
This example clearly demonstrates that the check can be as difficult and as time
consuming as the computation used to originally solve the equation. For this reason,
we tend to get lazy and not check our answers as we should. There is help, however,
as the graphing calculator can help us check the solutions of equations.
First, enter the solution 3/2 in your calculator screen, push the STO◮ button, then
push the X button, and execute the resulting command on the screen by pushing the
ENTER key. The result is shown in Figure 1(a).
Next, enter the expression 5/X+6/Xˆ2 and execute the resulting command on the
screen by pushing the ENTER key. The result is shown in Figure 1(b). Note that the
result is 6, the same as computed by hand above, and it matches the left-hand side of
the original equation (4). We’ve also used the calculator to check the second solution
x = −2/3. This is shown in Figure 4(c).
(a) (b) (c)
Figure 1. Using the graphing calculator to check the solutions of equation (4).
Let’s look at another example.
◮ Example 5. Solve the following equation for x.
2
x 2 = 1 − 2 x
First, multiply both sides of equation (6) by the common denominator x 2 .
(6)
Version: Fall 2007