Solve a Polynomial Inequality Graphically

Summary

This 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

• 📈 The video explains how to use a graph to solve polynomial inequalities.
• 🔍 For inequalities greater than or equal to zero, look for intervals where the graph is above the x-axis.
• 📌 When the inequality includes equality (e.g., greater than or equal to), include the x-intercepts in the solution.
• ➡️ Use interval notation to express the solution, starting from left to right on the graph.
• 🚫 For strict inequalities (e.g., greater than), do not include the x-intercepts.
• 🔢 If the inequality is less than zero, find the intervals where the graph is below the x-axis.
• 🔄 For less than inequalities, use parentheses to denote non-inclusive endpoints.
• 📋 In set notation, describe the intervals using 'less than' or 'greater than' language.
• 🔄 For inequalities less than or equal to zero, include the x-intercepts with brackets.
• 📉 The video provides examples of how to interpret the graph for both positive and negative y-values.
• ❓ 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

00:00

📈 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.

Keywords

💡Graph

A graph in the context of the video refers to a visual representation of a polynomial function, which helps in solving polynomial inequalities. It is used to identify intervals where the function's values are positive or negative. The video script mentions using a graph to find where the Y values are positive, indicating the function is greater than zero.

💡Polynomial Inequality

A polynomial inequality is an inequality that involves a polynomial expression. In the video, the focus is on solving inequalities that ask whether the polynomial is greater than or less than a certain value, including variations like 'greater than or equal to' and 'less than or equal to'. The script explains how to interpret these inequalities using a graph.

💡Interval Notation

Interval notation is a method of expressing a set of numbers on a number line. The video script discusses using interval notation to represent the solutions to the polynomial inequality, starting from the leftmost point to the rightmost point, and including or excluding endpoints based on the inequality's conditions.

💡Positive Values

Positive values in the context of the video refer to the Y values of the graph that are above the X-axis. The video explains that to find where the polynomial is greater than or equal to zero, one must look for intervals on the graph where Y is positive.

💡X-axis

The X-axis is a fundamental concept in graphing, representing the horizontal axis on which the values of X are plotted. The video script uses the X-axis to determine where the graph crosses or touches, indicating points of equality in the inequality.

💡Set Notation

Set notation is another way to express the solution set of an inequality. The video script suggests that while interval notation is more likely to be used, set notation could also be employed to describe the values of X that satisfy the inequality.

💡Brackets and Parentheses

Brackets and parentheses are used in interval notation to indicate whether an endpoint is included (inclusive) or excluded (exclusive) in the solution set. The video script explains that for 'greater than or equal to', a bracket is used to include the endpoint, whereas a parenthesis is used for 'greater than' to exclude the endpoint.

💡Y-values

Y-values in the video script refer to the vertical coordinates on the graph, representing the output of the polynomial function for a given X value. The script uses Y-values to determine the intervals where the polynomial is positive or negative.

💡Inequality Conditions

Inequality conditions are the relational operators used in mathematical inequalities, such as 'greater than', 'less than', 'greater than or equal to', and 'less than or equal to'. The video script explains how to interpret these conditions when solving polynomial inequalities using a graph.

💡Negative Values

Negative values in the video refer to the Y values of the graph that are below the X-axis. The script explains how to identify intervals where the polynomial is less than zero by looking at the portions of the graph that fall below the X-axis.

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