Express Inequalities as a Graph and Interval Notation

Summary

This video script offers a detailed tutorial on graphing and expressing inequalities in interval notation. It explains how to plot points for various inequalities, including less than, greater than, and less than or equal to, using open and closed circles to indicate inclusion or exclusion of endpoints. The script also demonstrates how to graph compound inequalities and provides examples of interval notation for each scenario, clarifying the use of parentheses and brackets for different cases.

Takeaways

• 📊 When graphing inequalities, use an open circle for values not included in the interval and a closed circle for values that are included.
• 🔍 For inequalities like 'X less than five', graph to the left of the number five, approaching negative infinity, and use parentheses to denote exclusion.
• 📌 Interval notation for 'X less than five' is written as "(-∞, 5)", indicating values less than five but not including five itself.
• 📈 For 'X greater than negative two', graph to the right of negative two, approaching positive infinity, and use parentheses to show exclusion of negative two.
• 🌐 The interval notation for 'X greater than negative two' is "(-2, ∞)", showing all values greater than negative two.
• 🔄 When an inequality includes an endpoint, like 'X less than or equal to negative one', use a square bracket to indicate inclusion.
• ⚠️ For 'X equals three', the interval notation can be expressed as "[3, 3]", showing that only the number three is included in the interval.
• 🔑 In interval notation, 'X greater than or equal to two' is written as "[2, ∞)", including two and extending to positive infinity.
• 🔄 For compound inequalities, like 'X greater than or equal to negative three and less than four', graph between the two values with appropriate inclusion or exclusion.
• 📐 The interval notation for the compound inequality 'X greater than or equal to negative three and less than four' is "[-3, 4)", including negative three but not four.
• 📉 For 'X greater than negative eight and less than or equal to zero', the interval does not include negative eight but does include zero, shown as "(-8, 0]" in interval notation.

Q & A

What does an open circle on a number on a graph represent in the context of inequalities?

An open circle on a number on a graph represents that the number is not included in the interval of the inequality.

How is the interval for the inequality 'X less than five' represented in interval notation?

The interval for 'X less than five' is represented in interval notation as '(-∞, 5)', indicating all values less than five, not including five itself.

What is the difference between a closed point and an open point on a graph when dealing with inequalities?

A closed point on a graph indicates that the value is included in the interval of the inequality, while an open point indicates that the value is not included.

How do you express the interval for 'X greater than negative two' in interval notation?

The interval for 'X greater than negative two' is expressed in interval notation as '(-2, ∞)', including all values greater than negative two, but not including negative two itself.

What is the interval notation for 'X less than or equal to negative one'?

The interval notation for 'X less than or equal to negative one' is '[-∞, -1]', including all values less than or equal to negative one.

How is the single value interval for 'X equals three' represented in interval notation?

The single value interval for 'X equals three' can be represented in interval notation as '[3, 3]', indicating that only the value three is included in the interval.

What does a closed point on a number signify when graphing the inequality 'X greater than or equal to two'?

A closed point on a number signifies that the number is included in the interval of the inequality, so for 'X greater than or equal to two', a closed point is made at two.

How do you express a compound inequality like 'negative three less than or equal to X less than four' in interval notation?

A compound inequality like 'negative three less than or equal to X less than four' is expressed in interval notation as '[-3, 4)', including negative three but not including four.

What is the interval notation for the inequality 'negative eight less than X less than or equal to zero'?

The interval notation for 'negative eight less than X less than or equal to zero' is '(-8, 0]', including zero but not including negative eight.

How do you determine whether to use a square bracket or a rounded parenthesis when expressing intervals involving infinity?

When expressing intervals involving infinity, a rounded parenthesis is always used. For finite endpoints that are included in the interval, a square bracket is used; if the endpoint is not included, a rounded parenthesis is used.

Can the interval notation for a single value be expressed using set notation?

Yes, the interval notation for a single value can also be expressed using set notation by placing the value within braces, such as '{3}' for the interval '[3, 3]'.

Outlines

00:00

📈 Graphing Inequalities and Interval Notation

This paragraph explains the process of graphing inequalities and expressing them in interval notation. It starts with the inequality X < 5, illustrating how to plot an open circle at 5 to indicate it's not included, and graphing to the left towards negative infinity. The interval notation for this is (-∞, 5). The explanation continues with X > -2, where an open circle is plotted at -2, and the graph extends to the right towards positive infinity, with the interval notation being (-2, ∞). The paragraph also covers inequalities including equality, such as X ≤ -1, which includes negative one, and is graphed with a closed circle, resulting in the interval notation (-∞, -1].

05:01

🔍 Special Cases in Inequality Graphing and Notation

The second paragraph delves into special cases of inequalities. It starts with X = 3, where the graph includes only the number 3, represented by a closed circle, and the interval notation is [3, 3]. Next, it discusses X ≥ 2, where the graph includes two and extends to positive infinity, with the interval notation [2, ∞). The paragraph then explains compound inequalities, such as -3 ≤ X < 4, which is graphed between -3 and 4, including -3 but not 4, resulting in the interval notation [-3, 4). Lastly, it covers -8 < X ≤ 0, where the graph is between -8 and 0, not including -8 but including 0, with the interval notation (-8, 0]. The paragraph emphasizes the use of open and closed circles to indicate whether endpoints are included in the interval.

Keywords

💡Inequality

Inequality refers to a mathematical expression that shows the relationship between two values that are not equal. In the context of the video, inequalities are used to define certain ranges of numbers. For example, 'X less than five' is an inequality that describes all values of X that are smaller than 5.

💡Interval Notation

Interval notation is a way of expressing a set of numbers that have a specific range, often used in mathematics to denote the set of all numbers between two points, including or excluding those points. The video explains how to express different types of inequalities in interval notation, such as '(-∞, 5)' for values less than five.

💡Open Point

An open point, or open circle, on a graph represents a number that is not included in the set of numbers defined by an inequality. The video uses the concept of an open point to illustrate that the number 5 is not part of the interval in the inequality 'X less than five'.

💡Closed Point

A closed point on a graph signifies that the particular number is included in the set of numbers defined by an inequality. In the script, a closed point is made on negative one for the inequality 'X less than or equal to negative one', indicating that negative one is part of the interval.

💡Negative Infinity

Negative infinity is a concept representing an infinitely large negative number. In the video, it is used to describe the lower bound of certain intervals, such as in 'X less than five', where the values approach negative infinity.

💡Positive Infinity

Positive infinity is the concept of an infinitely large positive number. The video mentions positive infinity when discussing intervals that extend indefinitely to the right, such as in the inequality 'X greater than negative two'.

💡Graph

In the context of the video, a graph is a visual representation of mathematical data, typically on a coordinate plane. The script describes how to graph inequalities by plotting points and drawing lines to represent the intervals defined by the inequalities.

💡Compound Inequality

A compound inequality is an inequality that combines two or more conditions. The video explains how to interpret and graph compound inequalities, such as 'negative three less than or equal to X less than four', which includes values between negative three and four, but not including four.

💡Set Notation

Set notation is a way of writing sets of numbers using curly braces and other symbols. The video briefly mentions set notation as an alternative way to express the interval for the inequality 'X equals three', using braces around the number three.

💡Square Bracket

In interval notation, a square bracket is used to indicate that an endpoint of the interval is included in the set. The video explains that a square bracket is used to the right of negative one in the interval notation for 'X less than or equal to negative one'.

💡Rounded Parenthesis

A rounded parenthesis in interval notation signifies that an endpoint of the interval is not included in the set. The video demonstrates this by using a rounded parenthesis to the right of five in the interval notation for 'X less than five'.

Highlights

Introduction to graphing inequalities and expressing intervals using interval notation.

Explanation of how to graph 'X less than five' with an open circle at five, indicating the value is not included in the interval.

Graphing values less than five extending to negative infinity for the inequality X < 5.

Expressing the interval for X < 5 using interval notation as (-∞, 5) with a rounded parenthesis to exclude five.

Graphing 'X greater than negative two' with an open circle at -2 and extending to positive infinity.

Using interval notation for X > -2 as (-2, ∞) with a rounded parenthesis to exclude -2.

Graphing 'X less than or equal to negative one' with a closed point at -1, including it in the interval.

Interval notation for X ≤ -1 as (-∞, -1], including -1 with a square bracket.

Graphing a single value 'X equals three' with a closed point at three.

Expressing the interval for X = 3 in interval notation as [3, 3], including the single value with square brackets.

Graphing 'X greater than or equal to two' with a closed point at 2 and extending to positive infinity.

Interval notation for X ≥ 2 as [2, ∞), including 2 with a square bracket and excluding infinity with a rounded parenthesis.

Understanding compound inequalities and graphing 'X greater than or equal to negative three and less than four'.

Interval notation for the compound inequality as [-3, 4), including -3 and excluding 4.

Graphing 'X greater than negative eight and less than or equal to zero' with open and closed points at -8 and 0, respectively.

Interval notation for 'X > -8 and X ≤ 0' as (-8, 0], excluding -8 and including 0.

Summary of using square brackets for included endpoints and rounded parentheses for excluded endpoints or infinity.

Final note on the importance of understanding interval notation and graphing techniques for inequalities.

Transcripts

In this exercise, we are given several inequalities

then asked to graph the interval

and then express the interval using interval notation.

Looking at the first inequality, we have X less than five.

Because inequality is less than, not less than

or equal to, five is not in the interval

which means we begin by graphing or plotting

an open point or open circle on five.

The open point indicates this value is not in the interval.

If it said less than or equal to, then we'd have

a closed point, indicating the value is in the interval.

And then because the inequality is X less than five,

values less than five are to the left.

We now graph to the left.

Notice how the values approach negative infinity.

This is the graph for the inequality X less than five.

Now let's express the interval using interval notation.

Because the values approach negative infinity

to the left and positive five to the right,

we begin with negative infinity comma five

and now need to decide whether to the right or left

we have a square bracket round parenthesis.

For positive or negative infinity,

we always have a rounded parenthesis

and then because interval does not include five,

we also have a rounded parenthesis to the right of five.

If it included five, we would have a square bracket

to the right of five.

For B, we have X greater than negative two.

Because the inequality is greater than not greater than

or equal two, negative two is not in the interval.

We begin by making an open point or open circle

on negative two.

This indicates negative two is not in the interval

and now because the inequality is X greater

than negative two, values are greater to negative two

to the right.

We graph to the right approaching positive infinity.

And now for the interval notation,

the values here are approaching negative two to the left

and positive infinity to the right.

We begin with negative two comma infinity.

Because negative two is not in the interval,

we have a rounded parenthesis to the left

and for positive and negative infinity,

we always have a rounded parenthesis.

This is the interval using interval notation.

Next, we have X less than or equal to negative one.

Because the inequality is less than or equal to negative one

negative one is in the interval,

we make a close point on negative one at this time

because negative one is in the interval.

The interval is X less than or equal to negative one.

The values are less than negative one to the left.

We graph to the left, approaching negative infinity,

which means for interval notation,

we begin with negative infinity comma negative one.

For negative infinity, we always have a rounded parenthesis

because the interval includes negative one,

we have a square bracket to the right of negative one.

So we can say the interval is closed on negative one for C

and for B, we can say the interval is open on negative two.

D is a little bit different.

D, we have X equals three

because the interval includes three,

we make a closed point on three

and this is the only number or only value in the interval,

so we are done with the graph.

It seems a little strange to express this

using interval notation,

but the interval begins and ends with three,

so one way to express this using interval notation

would be three comma three.

And because three is in the interval,

we would have a square bracket to the left of three

and to the right of three.

Another way to express this

using set notation would be to put a three in braces,

which would look like this.

For E, we have X greater than or equal to two.

The interval includes two.

We make a closed point on two.

We have X greater than or equal to two.

Values greater than two are to the right.

We graph to the right approaching positive infinity.

Using interval notation, the interval goes from two

and approaches infinity and includes two.

We have a square bracket to the left of two.

Positive infinity goes to the right forever

and therefore, it's never going to include infinity.

We make a rounded parenthesis to the right of infinity.

The last two are a comp of inequalities.

There's a couple ways to read this.

We can read it from left to right as negative three

less than or equal to X less than four,

but we can also start with the variable

and read this from right to left

as well as left to right.

We could say X is greater than or equal to negative three

and X is less than four.

Either way is correct.

Looking at the inequality symbol though,

notice how negative three is in the interval

and positive four is not, so we make a closed point

on negative three and an open point on positive four.

Again, we can read this as X is greater than or equal to

negative three, so we would graph to the right

of negative three.

But X is also less than four, so we graph to the left

of four, which means this compound inequality

is the interval between these two values,

including negative three and not including four.

To express this using interval notation,

we begin at negative three and we end at four.

The interval includes negative three.

We make a square bracket to the left of negative three.

It does not include four.

We use a rounded parenthesis to the right of four.

And finally for G, we can read this as negative eight

less than X, less than or equal to zero.

Or we can read this as X is greater than negative eight

and X is less than or equal to zero.

The interval does not include negative eight

because we read this X greater than negative eight.

We graph an open point on negative eight

and X is also less than or equal to zero.

Zero is in the interval, so we graph a closed point on zero

and we graph all the values between at negative eight

and zero.

This is the graph of the interval.

The interval is between negative eight and zero.

Negative eight is not in the interval.

We have a round parenthesis to the left of negative eight.

The interval does include zero and therefore,

we have a square bracket to the right of zero.

So again, if the end point is included,

we use a square bracket.

If it's not included or it's plus or minus infinity,

we always use a rounded parenthesis.

I hope you found this helpful.