1.11 Solving
Inequalities
by Graphing Calculator Methods
College
Algebra: One Step at a Time,
Pages 155 - 163: #5, 11, 23, 29
Pages 164 - 172: #17, Extra Problems
Dr. Robert J. Rapalje
Seminole Community College
Sanford, FL 32773
p. 161.
#5.
Solve for x. Give answers in interval notation.
.
Solution:
First, set the inequality to zero by adding
to
each side of the inequality:
,
and graph
with
a graphing calculator.
Since the
inequality is
,
you will be looking for values of x where the graph is
BELOW the x axis.
The graph in the standard window [-10,10]
of a graphing calculator should look like this:
In order to see the graph better, you may want to extend the y window of the
graph to [-10,20].
Notice that this graph is always above and
NEVER below
the x-axis. Therefore, there is
NO SOLUTION.
p. 162.
#11.
Solve for x. Give answers in interval notation.
.
Solution:
First, compare this exercise to #5 from page 163, which has already been
solved above. Except for the inequality sign, it is the same! As in the
previous exercise, by adding
to
each side of the inequality:
,
and graph
with
a graphing calculator.
The graph is exactly the same as in Exercise #5! However, since
the inequality is
,
you will be looking for values of x where the graph is
ABOVE the x axis.
The graph in the standard window [-10,10]
is given on the left, and an extended window for y from
[-10,20] is given on the right:
Notice that this graph is
ALWAYS ABOVE the x-axis. Therefore, the solution is
ALL REAL VALUES of x.
p. 163. #23.
Solve for
x. Give answers in interval notation.
.
Solution: First, set the inequality to zero by adding
to
each side of the inequality.
Graph with
a graphing calculator
.
When you enter this in the calculator, you will need double
parentheses. Be sure that you have parentheses around the numerator of the
fraction and also around the quantity that is within the absolute value.
When you enter the equation, the equation and the graph should look like
this in a standard window:
Since the inequality is
,
you will be looking for values of x, where the graph is
ON or ABOVE the x axis.
As you can see from the graph, there are two intervals in which the graph is
ABOVE the x-axis,
but you need to find the x-intercepts (zeros)
of the graph. You can do this with algebraic methods
(see “Algebraic
Method of Finding Endpoints” at the end of this
exercise!),
or you can use the “zeros method"
for the TI 83/84. (For the TI 85/86, it is called the “root
method"!--if you need the steps for TI 85/86
solution, just send me an Email !).
You can probably tell
that the right endpoint
is x=2, or you can
get it from the TABLE
function of the calculator. However, the
left endpoint is NOT that
obvious. For the TI 83/84, this is the method:
Type: [2nd]
[F4 (CALC)] [2 (zero)].
The calculator now
wants a “Left Bound”,
so give the calculator any value that is to the left of what you think the
x-intercept will
be. That is, give it -4
or -5
or -6,
and press [ENTER].
Next, the calculator wants a “Right Bound”,
so give it anything larger than this particular x-intercept, like
-2 or
-1 or 0.
Be sure that the number you choose is something less than 2. Then press
[ENTER]. The
calculator will then ask for a “Guess?”,
to which you can just press [ENTER]
again. Using this procedure, and using appropriate
Left and Right Bound
values for x = 2,
the calculator will give you the following
zeros (roots):
Of course,
.
Notice that this graph is always
ABOVE the x-axis when
x is to the left
of
or
to the right
of 
.
The graph on a number line, and the interval
notation, which
INCLUDES the ENDPOINTS,
should look like this:
Algebraic Method of Finding Endpoints
Weaknesses
of the graphing calculator method of solving problems include being sure
that the window of your calculator is large (or small!) enough to be sure
that you have ALL of the endpoints and being able to find their exact
value. Solving for the endpoints algebraically is often necessary to
overcome this. Besides, algebraic methods are often easier than calculator
methods. To find the endpoints for
,
simply change the inequality to an equation, and solve.
There are TWO solutions
for endpoints:
Multiply both sides of
each by the denominator which is 2:
With these as the endpoints, the solution from the graphing calculator is
ON or ABOVE the x-axis
as follows:
p. 163.
#29.
Solve for x. Give answers in interval notation.
.
Solution:
First, graph with a graphing calculator
. Since the inequality is
, you will be looking for values of
x where the graph is
above the x axis.
Notice
that this graph is always above the x-axis, except at the value of x=1. The
solution is therefore all values except 1. The graph on a number line and
the interval notation should look like this:
p. 168.
#17.
Solve for x. Give answers in
interval notation. 
.
Solution:
Start by setting the inequality to zero:
,
and graph
Notice that you don’t even need to multiply it out. If you do that’s fine,
but it’s not necessary. Since the
inequality is
,
you will be looking for values of x, where the graph is on or below the x axis.
The graph should look like this:
The graph
has roots at x=-4 and between 0 and 1. You can find the exact value of the
root either by algebra methods (set the equation equal to zero and solve for
x), or using the graphing calculator. For a TI 83 or TI 84, you can use [2nd]
[CALC] and then look for [2: zero] . Select a left bound of 0 or -1, and a
right bound of 1 or 2, and press [ENTER] [ENTER]. The calculator should
show the screen given above.
The
solution is the set of all values of x, for which the graph is on or below
the x axis. The graph on a number line and interval notation will be:
ALTERNATE
PROBLEM:
Solve for x. Give answers in interval notation.
.
Solution:
Start by setting the inequality to zero:
, and graph
Notice that you don’t even need to multiply it out. If you do that’s fine,
but it’s not necessary. Since the
inequality is
, you will be looking for values of x, where
the graph is on or below the x
axis. The graph should look like this:
The graph
has roots at x=0 and between -3 and -2. You can find the exact value of the
root either by algebra methods (set the equation equal to zero and solve for
x), or using the graphing calculator. For a TI 83 or TI 84, you can use [2nd]
[CALC] and then look for [2: zero] . Select a left bound of -3 or -4, and a
right bound of -1 or -2, and press [ENTER] [ENTER]. The calculator should
show the screen given above.
The
solution is the set of all values of x, for which the graph is on or below
the x axis. The graph on a number line and interval notation will be:
Extra Problem
(by Aimee).
Solve for
x. Give the answer in interval notation.
.
Solution:
Start by drawing the
graph of
Notice that the roots (or zeros) of this function are at x=8 and x=-5, and
there is an asymptote at x=3. Since the
inequality is
,
you will be looking for values of x, where the graph is
on or above the x axis.
The graph has roots at
x=-5 and 8. The vertical line at x=3 is not really a part of the graph, but
it is an asymptote, a line that the graph approaches by never actually
touches.
Since there are two
roots (zeros) and one asymptote, this gives you three endpoints on the
number line, and four intervals to consider for your solution. You must
select the intervals that are ON or ABOVE the x-axis.
In the
first interval, from
–infinity to -5, the graph is
below the x-axis.
In the
second interval,
from -5 to 3 the graph is
above the x-axis.
In the
third interval,
from 3 to 8, the graph is
below the x-axis.
In the
fourth interval,
from 8 to infinity, the graph is
above the x-axis.
Therefore the solution
consists of the second and fourth interval, where the graph is above the
x-axis, including the endpoints at x=-5 and x=8, since these are points that
are ON the x-axis.
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