Lesson Objectives

- Learn about inequality relationships between numbers
- Learn how to write a solution in interval notation
- Learn how to graph an interval on the number line
- Learn about set-builder notation

## What is an Inequality in Math?

An inequality in math is just a statement that two algebraic expressions are not equal in value. In our pre-algebra course, we learned about inequalities. We learned that as numbers move to the right on the number line, values increase, as numbers move to the left on the number line, values decrease. We can look at our number line above and note that (-5) is larger than (-7), since (-5) lies to the right of (-7) on the number line. We can express the relationship between the two numbers with an inequality symbol.

< : less than » the left number is less than the right number

> : greater than » the left number is greater than the right number

(-5) > (-7) » "read: -5 is greater than -7"

(-7) < (-5) » "read -7 is less than -5"

Notice how the correct symbol always points to the smaller number.

We will now introduce two new symbols. These symbols are:

≤ : less than or equal to » the left number is less than or equal to the right number

≥ : greater than or equal to » the left number is greater than or equal to the right number

The first two inequality symbols: "<", and ">" are known as strict inequalities. These do not allow for the possibility of equality. The second two inequalities: "≤", and "≥" are known as non-strict inequalities. These do allow for the possibility of equality.

Example 1: Replace the "?" with either "<" or ">" $$-6 \hspace{.5em}?\hspace{.5em}0$$ Since -6 is to the left of 0 on the number line, it is a smaller number. We will use a less than symbol, which will point toward -6. $$-6 < 0$$

x > -2

This inequality is satisfied when x is any value that is larger than (-2). (-2) itself is not a solution since (-2) is not greater than (-2), they are equal. Remember numbers increase as we move to the right on the number line. Therefore, any value to the right of (-2) on the number line is a solution. We can show this graphically by shading all values to the right of (-2) on the number line. Since (-2) is not a solution here, we exclude it by placing an open circle at (-2). Notice how our right arrow is also shaded. This indicates that all numbers are shaded out to positive infinity.

Another way to show this is with a special type of notation known as interval notation. Interval notation allows us to show a solution as a range of values. We write a solution in interval notation using the following rules:

smallest value, largest value

The smallest value is on the left, followed by a comma, and the largest value is on the right. In many cases, we can't actually list the smallest or largest value, that's where certain symbols come into play.

A "(" left parenthesis indicates that the smallest value is anything larger than the smallest value given.

A "[" left bracket indicates that the smallest value is the smallest value given.

A ")" right parenthesis indicates that the largest value is anything smaller than the largest value given.

A "]" right bracket indicates that the largest value is the largest value given.

This may seem a bit confusing at first, but with some practice, it becomes very easy to understand.

With our inequality of x > -2, we can notate this as:

(-2, ∞)

We use a left parenthesis here since x can't be (-2), but anything larger works. This means (-1.999) works as a solution as would (-1.9999). Since we can keep creating numbers closer to (-2) that are larger than (-2), we just notate this with our parenthesis and say anything larger than (-2). For the largest value, we put the infinity symbol "∞". Infinity is just a concept in math that is used when the value isn't countable. This is because there is no largest value, x can be anything larger than (-2). When we use either positive or negative infinity, we always use a parenthesis. Let's take a look at a few examples.

Example 2: Write each interval using interval notation

x < 4

Since x is strictly less than 4, we place a 4 in the spot of the largest value and a right parenthesis next to it. This indicates that 4 is not included:

, 4)

On the left, there isn't a smallest value, x can be anything less than 4. We can notate this using negative infinity. A parenthesis is always used with infinity.

(-∞, 4)

Example 3: Write each interval using interval notation

x ≥ -3

Since x can be (-3) or anything larger, we place (-3) in the spot of the smallest value and a left bracket next to it. This indicates that (-3) is included:

[-3,

On the right, there isn't a largest value, x can be anything larger than (-3). We can notate this with positive infinity. A parenthesis is always used with infinity.

[-3, ∞)

Example 4: Write each interval using interval notation

-1 < x ≤ 5

In this case, x can be any value that is larger than (-1) up to and including 5. We see that x has to be strictly greater than (-1), this means we will use (-1) as the smallest value and place a parenthesis next to it:

(-1,

For the largest value, 5 is included. This is from the non-strict inequality "≤". We will use a right bracket next to 5:

(-1, 5]

A parenthesis or open circle implies the number is not included:

Suppose we had the following inequality:

x > 3

Using an open circle: Using a parenthesis: A bracket or filled-in circle implies that the number is included:

Suppose we had the following inequality:

x ≤ 3

Using a filled-in circle: Using a bracket: Let's look at a few examples.

Example 5: Graph each inequality on the number line

x ≤ 1

Place a right bracket or filled-in circle at 1. We then shade everything to the left. Example 6: Graph each inequality on the number line

-5 ≤ x < 4

Place a left bracket or filled-in circle at (-5). Place a right parenthesis or open circle at 4. We then shade everything between the two numbers (-5 and 4).

{x | x has some property}

The first part is read as: "the set of all x"

"|" is read as: "such that"

The last part is read as: "x has some property"

For right now, when we hear "the set of all x", we are saying x is some real number. We then become more specific by saying "such that" and lastly we give our condition. So x is a real number such that it has a specific property.

Let's say we had the problem:

x > 14

We can write this in set-builder notation as:

{x | x > 14}

Basically, this is just saying that x is any real number that is larger than 14. Let's look at a few examples that deal with interval notation, graphing intervals, and set-builder notation.

Example 7: Write each in interval notation, set-builder notation, and graph the interval

x > 9

Interval Notation:

(9, ∞)

Set-Builder Notation:

{x | x > 9}

Graphing the Interval: Example 8: Write each in interval notation, set-builder notation, and graph the interval

-4 ≤ x < 12

Interval Notation:

[-4, 12)

Set-Builder Notation:

{x | -4 ≤ x < 12}

Graphing the Interval: Example 9: Write each in interval notation, set-builder notation, and graph the interval

-19 ≤ x ≤ -1

Interval Notation:

[-19, -1]

Set-Builder Notation:

{x | -19 ≤ x ≤ -1}

Graphing the Interval:

< : less than » the left number is less than the right number

> : greater than » the left number is greater than the right number

(-5) > (-7) » "read: -5 is greater than -7"

(-7) < (-5) » "read -7 is less than -5"

Notice how the correct symbol always points to the smaller number.

We will now introduce two new symbols. These symbols are:

≤ : less than or equal to » the left number is less than or equal to the right number

≥ : greater than or equal to » the left number is greater than or equal to the right number

The first two inequality symbols: "<", and ">" are known as strict inequalities. These do not allow for the possibility of equality. The second two inequalities: "≤", and "≥" are known as non-strict inequalities. These do allow for the possibility of equality.

Example 1: Replace the "?" with either "<" or ">" $$-6 \hspace{.5em}?\hspace{.5em}0$$ Since -6 is to the left of 0 on the number line, it is a smaller number. We will use a less than symbol, which will point toward -6. $$-6 < 0$$

### Interval Notation

When we work with linear equations in one variable, in most cases we have one single value as a solution. Suppose we look at the following simple equation: $$2x - 5=-9$$ We can quickly solve this equation: $$2x - 5=-9$$ $$2x=-4$$ $$\frac{2}{2}x=\frac{-4}{2}$$ $$x=-2$$ For this equation to be true, x can only be (-2). We can show this graphically as a dot or filled-in circle placed at (-2) on the number line. When we work with inequalities, we generally have more than one value which will work as a solution. Consider the following simple inequality:x > -2

This inequality is satisfied when x is any value that is larger than (-2). (-2) itself is not a solution since (-2) is not greater than (-2), they are equal. Remember numbers increase as we move to the right on the number line. Therefore, any value to the right of (-2) on the number line is a solution. We can show this graphically by shading all values to the right of (-2) on the number line. Since (-2) is not a solution here, we exclude it by placing an open circle at (-2). Notice how our right arrow is also shaded. This indicates that all numbers are shaded out to positive infinity.

Another way to show this is with a special type of notation known as interval notation. Interval notation allows us to show a solution as a range of values. We write a solution in interval notation using the following rules:

smallest value, largest value

The smallest value is on the left, followed by a comma, and the largest value is on the right. In many cases, we can't actually list the smallest or largest value, that's where certain symbols come into play.

A "(" left parenthesis indicates that the smallest value is anything larger than the smallest value given.

A "[" left bracket indicates that the smallest value is the smallest value given.

A ")" right parenthesis indicates that the largest value is anything smaller than the largest value given.

A "]" right bracket indicates that the largest value is the largest value given.

This may seem a bit confusing at first, but with some practice, it becomes very easy to understand.

With our inequality of x > -2, we can notate this as:

(-2, ∞)

We use a left parenthesis here since x can't be (-2), but anything larger works. This means (-1.999) works as a solution as would (-1.9999). Since we can keep creating numbers closer to (-2) that are larger than (-2), we just notate this with our parenthesis and say anything larger than (-2). For the largest value, we put the infinity symbol "∞". Infinity is just a concept in math that is used when the value isn't countable. This is because there is no largest value, x can be anything larger than (-2). When we use either positive or negative infinity, we always use a parenthesis. Let's take a look at a few examples.

Example 2: Write each interval using interval notation

x < 4

Since x is strictly less than 4, we place a 4 in the spot of the largest value and a right parenthesis next to it. This indicates that 4 is not included:

, 4)

On the left, there isn't a smallest value, x can be anything less than 4. We can notate this using negative infinity. A parenthesis is always used with infinity.

(-∞, 4)

Example 3: Write each interval using interval notation

x ≥ -3

Since x can be (-3) or anything larger, we place (-3) in the spot of the smallest value and a left bracket next to it. This indicates that (-3) is included:

[-3,

On the right, there isn't a largest value, x can be anything larger than (-3). We can notate this with positive infinity. A parenthesis is always used with infinity.

[-3, ∞)

Example 4: Write each interval using interval notation

-1 < x ≤ 5

In this case, x can be any value that is larger than (-1) up to and including 5. We see that x has to be strictly greater than (-1), this means we will use (-1) as the smallest value and place a parenthesis next to it:

(-1,

For the largest value, 5 is included. This is from the non-strict inequality "≤". We will use a right bracket next to 5:

(-1, 5]

### Graphing Intervals on the Number Line

Once we have learned interval notation, graphing an interval on a number line is very simple. For a greater than, we shade everything to the right of the smallest value, and for a less than, we shade everything to the left of the largest value. The one difference between textbooks seems to be the use of circles, parentheses, and brackets.A parenthesis or open circle implies the number is not included:

Suppose we had the following inequality:

x > 3

Using an open circle: Using a parenthesis: A bracket or filled-in circle implies that the number is included:

Suppose we had the following inequality:

x ≤ 3

Using a filled-in circle: Using a bracket: Let's look at a few examples.

Example 5: Graph each inequality on the number line

x ≤ 1

Place a right bracket or filled-in circle at 1. We then shade everything to the left. Example 6: Graph each inequality on the number line

-5 ≤ x < 4

Place a left bracket or filled-in circle at (-5). Place a right parenthesis or open circle at 4. We then shade everything between the two numbers (-5 and 4).

### Set-Builder Notation

Lastly, we want to look at set-builder notation. This type of notation comes up pretty often in college algebra and higher-level math. Essentially set-builder notation looks like:{x | x has some property}

The first part is read as: "the set of all x"

"|" is read as: "such that"

The last part is read as: "x has some property"

For right now, when we hear "the set of all x", we are saying x is some real number. We then become more specific by saying "such that" and lastly we give our condition. So x is a real number such that it has a specific property.

Let's say we had the problem:

x > 14

We can write this in set-builder notation as:

{x | x > 14}

Basically, this is just saying that x is any real number that is larger than 14. Let's look at a few examples that deal with interval notation, graphing intervals, and set-builder notation.

Example 7: Write each in interval notation, set-builder notation, and graph the interval

x > 9

Interval Notation:

(9, ∞)

Set-Builder Notation:

{x | x > 9}

Graphing the Interval: Example 8: Write each in interval notation, set-builder notation, and graph the interval

-4 ≤ x < 12

Interval Notation:

[-4, 12)

Set-Builder Notation:

{x | -4 ≤ x < 12}

Graphing the Interval: Example 9: Write each in interval notation, set-builder notation, and graph the interval

-19 ≤ x ≤ -1

Interval Notation:

[-19, -1]

Set-Builder Notation:

{x | -19 ≤ x ≤ -1}

Graphing the Interval:

#### Skills Check:

Example #1

Write -3 < x ≤ 9 in set-builder notation

Please choose the best answer.

A

$$\{x | (-3, 9)\}$$

B

$$\{x | x > -3 ∪ x≤ 9\}$$

C

$$\{x |x=> [-3, 9]\}$$

D

$$\{x | -3 < x ≤ 9\}$$

E

$$(x=> -3, x=> 9]$$

Example #2

Write x > 7 in interval notation

Please choose the best answer.

A

$$(7, \infty)$$

B

$$[7, \infty]$$

C

$$[7, \infty)$$

D

$$[-\infty, 7]$$

E

$$(-\infty, 7)$$

Example #3

Write x ≤ -3 in interval notation

Please choose the best answer.

A

$$(-\infty, -3)$$

B

$$(-\infty, -3]$$

C

$$(-3, \infty)$$

D

$$[-3, \infty]$$

E

$$[-3, \infty)$$

Example #4

Write -6 < x ≤ 7 in interval notation

Please choose the best answer.

A

$$(-6, 7)$$

B

$$(-\infty, -6) ∪ [7, \infty)$$

C

$$(-6, 7]$$

D

$$[-\infty, -6] ∪ [7, \infty]$$

E

$$[-6, 7]$$

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