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3.06 Polynomial and Fractional Inequalities by Graphing Calculator Methods College Algebra: One Step at a Time, Pages 413 - 424: 29, 3 Extra Problems Dr. Robert J. Rapalje Seminole State College of Florida Sanford, FL 32773
P. 423 #29. Solve for x. (Give the answer in interval notation.)
Solution: First, you must set the inequality to zero by subtracting 1 from each side of the inequality.
Next, draw the graph of
Notice that the roots (or zeros) of this function are at x=-2 and x=6, and
there are asymptotes at x=0 and x=4. These zeros and asymptotes can and
will be obtained algebraically. Since the
inequality is
Since there are two roots (zeros) and two asymptotes, this gives you four endpoints on the number line, and five intervals to consider for your solution. You must select the intervals that are ABOVE the x-axis.
In the first interval, from –infinity to -2, the graph is below the x-axis. In the second interval, from -2 to 0 the graph is above the x-axis. In the third interval, from 0 to 4, the graph is below the x-axis. In the fourth interval, from 4 to 6, the graph is above the x-axis. In the fifth interval, from 6 to infinity, the graph is below the x-axis.
Therefore the solution consists of the second and fourth interval, where the graph is above the x-axis (endpoints are NOT included in this one!).
FINAL ANSWER:
— Finding the Roots and Asymptotes Algebraically — 1. Asymptotes--Set denominators ¹ 0.
Solve
2. Roots--Change the inequality to an equation and solve for x.
Solve
Extra Problem 1 (By Vera). Solve for x. (Give the answer in interval notation.)
Solution:
First, you must set the inequality to zero by subtracting
Next, find the endpoints by I. finding any values of y that make a denominator zero, and
II.
solving the EQUATION
I. Solve
II. Solve
So the endpoints are Next, draw the graph of
Since the
inequality is
In the first interval,
from –infinity
to
In the second interval,
from
In the third interval, from 2 to infinity, the graph is below the x-axis.
Therefore the solution consists of the first and third interval, where the graph is below the x-axis (endpoints are NOT included!).
Extra Problem 2 (By Vera). Solve for x. (Give the answer in interval notation.)
Solution:
First, you must set the inequality to zero by subtracting
Next, find the endpoints by I. finding any values of y that make a denominator zero, and
II.
solving the EQUATION
I. Solve
II. Solve
So the endpoints are Next, draw the graph of
Since the
inequality is
In the
first interval, from
–infinity to
In the second interval,
from
In the third interval,
from
Therefore the solution consists of the first and third interval, where the graph is below the x-axis (endpoints are NOT included!).
p. 163. #29. 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
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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