Solve a Polynomial Inequality Graphically
Summary
TLDRThis video tutorial demonstrates how to solve polynomial inequalities using a graph. It explains that for inequalities like 'greater than or equal to zero,' one should identify intervals where the graph is above the x-axis, including points where it crosses the axis. The video also covers 'less than zero' scenarios, focusing on areas below the x-axis, excluding crossing points. Interval notation is emphasized, with examples provided for both inclusive and exclusive intervals, and set notation is briefly discussed for completeness.
Takeaways
- đ The video explains how to use a graph to solve polynomial inequalities.
- đ For inequalities greater than or equal to zero, look for intervals where the graph is above the x-axis.
- đ When the inequality includes equality (e.g., greater than or equal to), include the x-intercepts in the solution.
- âĄïž Use interval notation to express the solution, starting from left to right on the graph.
- đ« For strict inequalities (e.g., greater than), do not include the x-intercepts.
- đą If the inequality is less than zero, find the intervals where the graph is below the x-axis.
- đ For less than inequalities, use parentheses to denote non-inclusive endpoints.
- đ In set notation, describe the intervals using 'less than' or 'greater than' language.
- đ For inequalities less than or equal to zero, include the x-intercepts with brackets.
- đ The video provides examples of how to interpret the graph for both positive and negative y-values.
- â The video encourages viewers to ask questions or request coverage of other topics.
Q & A
What is the main topic of the video?
-The main topic of the video is using a graph to solve polynomial inequalities.
What are the two scenarios discussed in the video for solving polynomial inequalities?
-The two scenarios discussed are finding when the polynomial is greater than or less than zero.
What does 'greater than or equal to zero' signify in the context of polynomial inequalities?
-In the context of polynomial inequalities, 'greater than or equal to zero' signifies looking for all x values where y is positive, including where the graph crosses the x-axis.
How does the graph help in identifying the intervals for 'greater than or equal to zero'?
-The graph helps by showing which portions of the graph are above the x-axis, indicating positive y-values, and including the points where the graph crosses the x-axis.
What is the significance of the x-axis in solving 'greater than or equal to zero' inequalities?
-The x-axis is significant because it represents the points where y equals zero, and these points are included in the solution set for 'greater than or equal to zero' inequalities.
What is the interval notation for the values of x that make y positive, according to the video?
-The interval notation includes all x values from negative infinity to negative 1, and from 1 to 3, using brackets to include the endpoints where the graph crosses the x-axis.
How does the video explain the process for finding intervals where y is less than zero?
-The video explains that for y values less than zero, one should look at the portions of the graph below the x-axis and use parentheses to indicate non-inclusive endpoints.
What is the difference between using brackets and parentheses in interval notation as per the video?
-Brackets are used to include the endpoint (for 'greater than or equal to' scenarios), while parentheses are used to exclude the endpoint (for 'less than' scenarios).
How does the video suggest representing the solution set in set notation?
-The video suggests representing the solution set in set notation by stating the set of all x values that satisfy the inequality, including or excluding the endpoints as appropriate.
What advice does the video give for viewers who have questions or need further topics covered?
-The video encourages viewers to ask questions or request additional topics for future videos.
Can you provide an example of how to interpret the graph for a 'less than zero' scenario?
-For a 'less than zero' scenario, the video suggests looking at the intervals from negative 1 to 1 (not inclusive) and from 3 to infinity, as these intervals yield negative y-values.
Outlines
đ Solving Polynomial Inequalities with Graphs
This video tutorial explains how to use a graph to solve polynomial inequalities. The focus is on identifying intervals where the polynomial is greater than or equal to zero. The presenter explains that positive Y values are sought, which correspond to portions of the graph above the X-axis. The process involves looking for intervals on the graph where the polynomial yields positive results. The video also discusses how to use interval notation to express these intervals, starting from negative infinity and including points where the graph crosses the X-axis. The presenter emphasizes that for greater than or equal to zero, the points where the graph intersects the X-axis are included, using brackets to denote this inclusion. Additionally, the video touches on how to express the solution in set notation, indicating the set of all X values that satisfy the inequality.
Mindmap
Keywords
đĄGraph
đĄPolynomial Inequality
đĄInterval Notation
đĄPositive Values
đĄX-axis
đĄSet Notation
đĄBrackets and Parentheses
đĄY-values
đĄInequality Conditions
đĄNegative Values
Highlights
Introduction to using a graph to solve a polynomial inequality.
Explanation of two different scenarios for polynomial inequalities: greater than or less than.
Description of how to interpret 'greater than or equal to zero' on a graph.
Identification of intervals on the graph where Y is positive.
Guidance on using interval notation for solutions that include the x-axis.
Clarification on including points where the graph crosses the x-axis.
Explanation of how to find intervals for 'less than 0' on the graph.
Identification of intervals where Y values are negative.
Instructions on using parentheses in interval notation for exclusive intervals.
Conversion of interval notation to set notation for a 'less than 0' scenario.
Discussion on the inclusion of endpoints in set notation for 'less than or equal to'.
Summary of the process for solving polynomial inequalities using a graph.
Encouragement for viewers to ask questions or request additional topics.
Transcripts
hi for this video what we're going to do
is we're gonna use a graph to help solve
a polynomial inequality so with this one
basically what you're looking for I have
two different scenarios because these
are the two things that you could be
asked whether it's going to be greater
than or less than it could be greater
than or equal to it could be less than
or equal to it could be just greater
than but for all of them you basically
are doing the same thing with this
greater than or equal to zero means
we're looking for all X values where Y
is positive so we're looking for
positive values in Y so if we look at
our graph we can see that from this
portion right here and this portion
right here are both above the x axis
which means that they have positive Y
outputs so that's what we're looking for
so when you have a graph you simply are
just looking for the intervals if you're
using interval notation you always start
from left to right and you're just going
to report the X values since this is
greater than or equal to that means that
it includes 0 so it includes the point
where it crosses the x axis so for this
one it does include the x axis if it was
just greater than it would not include
those points so starting from left to
right we can see that starting at
negative infinity all the way up to
negative 1 on the x axis that yields a
y-value that is positive so we would
start with negative infinity and all the
way from negative infinity until
negative 1 our values are above the
graph so at negative 1 it does switch
because it includes it we would use a
bracket and we do have another interval
where this also occurs so from 1 to 3 is
also the X values that would make Y
positive so if I plugged in any number
from 1 to 3 into the equation of this
graph it would give me an output that is
if you're working in set notation I
would say more likely interval notation
is probably going to be used but if you
are working in set notation you would
just say the set of all values of X such
that X is less than or equal to negative
1 less than or equal to negative 1 just
means below negative 1 or from 1 to 3
where it's included so it includes both
1 and 3 so for our second one we're now
looking for all values where our Y
values are less than 0 so now we're
looking at these values down here that
fall below the x axis so we can see and
this time it does not include the 0
because it's just lessen so we're
looking only for the negative outputs so
we can see that from negative 1 to 1 not
inclusive so that means we would use a
parenthesis so from negative 1 to 1 it
would yield a negative output and then
any values over 3 so from 3 to infinity
would yield an output that is less than
0 had this been less than or equal to we
would have used a hard bracket at the 3
and then around both of these 4 set
notation just in case you need to use
set notation you would just write it as
X such that X is between negative 1 and
1
it's not inclusive so we would not put
the equal to and then for all values of
X that are greater than 3 as always
thanks for watching if you have any
questions or have some other topics that
you need me to cover please just let me
know
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