Second Edition: 2012-2013

7.4.
SOLVING RATIONAL EQUATIONS
451
23.
The least common denominator (LCD) is
x
2
, so first clear fractions by
multiplying both sides of the equation by the LCD.
2
x
= 3 +
44
x
Original equation.
x
[2
x
] =
3 +
44
x
x
Multiply both sides by
x
.
x
[2
x
] =
x
[3] +
x
44
x
Distribute
x
.
2
x
2
= 3
x
+ 44
Cancel and simplify.
The resulting equation is nonlinear (
x
is raised to a power larger than 1). Make
one side zero
2
x
2
−
3
x
= 44
Subtract 3
x
from both sides.
2
x
2
−
3
x
−
44 = 0
Subtract 44 from both sides.
Compare 2
x
2
−
3
x
−
44 with
ax
2
+
bx
+
c
and note that
ac
= (2)(
−
44) and
b
=
−
3. The integer pair
−
11 and 8 have product
ac
=
−
88 and sum
b
=
−
3.
Replace the middle term with a combination of like terms using this pair, then
factor by grouping.
2
x
2
−
11
x
+ 8
x
−
44 = 0
−
3
x
=
−
11
x
+ 8
x
x
(2
x
−
11) + 4(2
x
−
11) = 0
Factor by grouping.
(
x
+ 4)(2
x
−
11) = 0
Factor out 2
x
−
11.
Use the zero product property to complete the solution. Either the first factor
is zero or the second factor is zero.
x
+ 4 = 0
or
2
x
−
11 = 0
x
=
−
4
x
=
11
2
Hence, the solutions are
x
=
−
4 and
x
= 11
/
2.
Graphical solution:
Make one side of the equation zero.
2
x
−
3
−
44
x
= 0
Load the left-hand side of the equation into
Y1
as
2*X-3-44/X
(see image
on the left), then select
6:ZStandard
from the ZOOM menu to produce the
image at the right.
Second Edition: 2012-2013

452
CHAPTER 7.
RATIONAL FUNCTIONS
Next, the solutions of
2
x
−
3
−
44
x
= 0
are found by noting where the graph of
y
= 2
x
−
3
−
44
/x
cross the
x
-axis. Select
2:zero
from the CALC menu. Use the arrow keys to move the cursor to the
left of the first
x
-intercept, then press ENTER to set the “Left bound.” Next,
move the cursor to the right of the first
x
-intercept, then press ENTER to set
the “Right bound.” Finally, leave the cursor where it is and press ENTER to
set your “Guess.” The calculator responds with the result shown in the figure
on the left.
Repeat the zero-finding procedure to capture the coordinates of
the second
x
-intercept (see the image on the right).
Reporting the solution on your homework.
Second Edition: 2012-2013