Lesson Objectives

- Demonstrate an understanding of how to solve a Linear Equation in One Variable
- Demonstrate an understanding of the inequality symbols "<", ">", "≤", and "≥"
- Learn how to notate an interval using interval notation
- Learn how to graph an interval on the number line
- Learn how to solve an inequality using the Addition Property of Inequality
- Learn how to solve an inequality using the Multiplication Property of Inequality

## How to Solve a Linear Inequality in One Variable

### What is an Inequality?

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.

### Interval Notation

When we worked with equations, in most cases we had 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

2/2 x = -4/2

x = -2

For this equation to be true, x can only be (-2). We can show this graphically as a dot placed at (-2) on the number line. When we work with inequalities, we generally have more than one value that 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 1: 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 2: 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 3: 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 4: 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 5: 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).

### Addition Property of Inequality

Once we have learned interval notation and how to graph an interval on the number line, it is time to move into solving a linear inequality in one variable. We will begin with a simple inequality of the form:x + 5 < 7

How can we solve this type of inequality? Recall with equations, we learned about the addition property of equality. This allows us to add or subtract the same value to or from both sides of an equation without changing the solution. The same property exists for inequalities. The addition property of inequality allows us to add or subtract the same value to or from both sides of an inequality without changing the solution. For our inequality, we want to isolate x and we can do this by subtracting away 5 from each side of the inequality.

x + 5 < 7

x + 5 - 5 < 7 - 5

x < 2

Checking an inequality is a bit more involved. We first replace our inequality symbols with an equality and check:

x + 5 = 7

x = 2

2 + 5 = 7

7 = 7 (true)

After this is confirmed, we try a value that matches with our inequality solution. Here x < 2, so we try a value less than 2. We can plug in a 1 for x and see what happens:

1 + 5 < 7

6 < 7 (true)

Lastly, we can check a value on the other side of 2, a value larger that shouldn't work. Let's plug in a 3 for x and see what happens:

3 + 5 < 7

8 < 7 (false)

We can see a value that shouldn't work, didn't work. It produced a false result, this tells us our answer x < 2 is correct. Since it is so time-consuming, we will omit the check for inequalities moving forward.

Let's try a few examples.

Example 6: Solve each inequality, graph, write in interval notation.

x - 11 < 7

We can isolate x by adding 11 to both sides of the inequality:

x - 11 + 11 < 7 + 11

x < 18

Interval Notation:

(-∞, 18)

Graphing the Interval: Example 7: Solve each inequality, graph, write in interval notation.

x + 3 ≥ 9

We can isolate x by subtracting 3 from both sides of the inequality:

x + 3 - 3 ≥ 9 - 3

x ≥ 6

Interval Notation:

[6, ∞)

Graphing the Interval:

### Multiplication Property of Inequality

When we solved equations, we learned about the multiplication property of equality. This property allows us to multiply or divide both sides of an equation by the same non-zero number. Suppose we run across an inequality such as:3x < 15

How can we solve this inequality? We can use the multiplication property of inequality. This property is slightly different than its equation counterpart. The multiplication property of inequality allows us to multiply or divide both sides of an inequality by the same positive number and not change the solution. Notice the positive part in the definition. Let's look at our example:

3x < 15

We can isolate x by dividing both sides by 3:

3/3 x < 15/3

x < 5

Now what if we had an inequality such as:

-3x < 15

Let's try the same procedure and divide both sides by (-3) to isolate x:

-3/-3 x < 15/-3

x < -5

If we check this solution, we will see that it is not valid. Let's try to plug in a number less than -5 for x and see if we get a true statement:

Plug in a -6 for x in -3x < 15

-3(-6) < 15

18 < 15 (false)

Why is our result not valid? The second part of the multiplication property of inequality tells us if we multiply or divide by a negative number, we must flip the direction of the inequality symbol. So in our case of x < -5, we would need to change the < part to >. So our solution becomes:

x > -5

Now let's try a value that's greater than -5, like -4:

-3(-4) < 15

12 < 15 (true)

This is a property that is often overlooked. It is something that requires practice to remember. Let's take a look at a few examples.

Example 8: Solve each inequality, graph, write in interval notation.

9x ≥ 27

We can divide each side by 9 to isolate x:

9/9 x ≥ 27/9

x ≥ 3

Interval Notation: [3, ∞)

Graphing the Interval: Example 9: Solve each inequality, graph, write in interval notation.

$$-\frac{1}{2}x < -7$$ Multiply each side by (-2), remember if we multiply by a negative we must flip the inequality symbol: $$-\frac{1}{2}x \cdot -2 > -7 \cdot -2$$ $$\require{cancel}\cancel{-}\frac{1}{\cancel{2}}x \cdot \cancel{-2}> 14$$ $$x > 14$$ Interval Notation:

(14, ∞)

Graphing the Interval:

#### Skills Check:

Example #1

Write in interval notation. $$-3 < x ≤ 11$$

Please choose the best answer.

A

$$(-3, 11)$$

B

$$(-3, 11]$$

C

$$[11, -3)$$

D

$$(-3, \infty)$$

E

$$(-\infty, 11]$$

Example #2

Solve the inequality. Write the answer in interval notation. $$x - 7 ≥ 10$$

Please choose the best answer.

A

$$(3, \infty)$$

B

$$(7, \infty)$$

C

$$[17, \infty)$$

D

$$(-\infty, 17]$$

E

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

Example #3

Solve the inequality. Write the answer in interval notation. $$-9x ≥ 405$$

Please choose the best answer.

A

$$\left[\frac{3}{4}, \infty\right)$$

B

$$\left[45 , \infty\right)$$

C

$$\left(-\infty , -45\right]$$

D

$$\left(-\infty , \frac{3}{4}\right]$$

E

$$\left(\frac{4}{3}, \infty\right)$$

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