Solving a Rational EquationSolve
Multiply.
Standard form
Factor.or Zero-factor property
or Proposed solutions
Because is not in the domain, it is not a solution. Check that the solution set
is E NOW TRY
12 F.
-
1
3x=
1
2
x=-
1
3
3 x+ 1 = 0 2 x- 1 = 0
13 x+ 1212 x- 12 = 0
6 x^2 - x- 1 = 0
2 x= 3 x+ 1 - 6 x^2
x 13 x+ 12 a
2
3 x+ 1
b = x 13 x+ 12 a
1
x
b -x 13 x+ 12 a
6 x
3 x+ 1
b
x 13 x+ 12 a
2
3 x+ 1
b = x 13 x+ 12 a
1
x
-
6 x
3 x+ 1
b
2
3 x+ 1
=
1
x
-
6 x
3 x+ 1
.
EXAMPLE 5
SECTION 7.4 Equations with Rational Expressions and Graphs 389
NOW TRY
EXERCISE 5
Solve.
2 x
x- 2=
- 3
x+
4
x- (^2) Multiply by the
LCD, x 13 x+ 12.
Distributive property
OBJECTIVE 3 Recognize the graph of a rational function. A function de-
fined by a quotient of polynomials is a rational function.Because one or more
values of xmay be excluded from the domain of a rational function, their graphs are
often discontinuous.That is, there will be one or more breaks in the graph.
One simple rational function, defined by and graphed in FIGURE 2, is
the reciprocal function.The domain of this function includes all real numbers
except 0. Thus, this function pairs every real number except 0 with its reciprocal.
ƒ 1 x 2 =^1 x
xy- 3 – 2 – 1
- 1
- 2
- 3
23
1
032f (x)^1 x
Vertical
Asymptote
x 0Horizontal
Asymptote
y 0FIGURE 2The closer negative values of xare to 0,
the smaller (“more negative”) yis.The closer positive values of
xare to 0, the larger yis.Reciprocal functionDomain:
Range: 5 x|xZ 065 x|xZ 06ƒ 1 x 2
1
xSince the domain of this function includes all real numbers except 0, there is no
point on the graph with The vertical line with equation is called a
vertical asymptoteof the graph. Also, the horizontal line with equation is
called a horizontal asymptote.
y= 0
x=0. x= 0
x 0.1 0.25 0.5 1 2 3
y - 31 - 21 - 1 - 2 - 4 - 10 10 4 2 1 12 31- 3 - 2 - 1 - 0.5 -0.25 -0.1
NOW TRY ANSWER
- E-^32 F
We must exclude
and 0 from the domain.-^13