### Are you looking forward to driving a car?

The law states that you need to be at least 16 years old to drive a car.

The law doesn't say you have to be exactly 16.

In math, an exact amount comes from an equation — but if the amount has a range of possible values, this is called an inequality.

There are inequalities everywhere in the real world. Mathematics is just a way to model and describe the real world.

### Can you write this inequality mathematically?

We can say that, x = the age at which you can drive a car.

Then x is greater (>) or equal (=) to 16: x ≥ 16

## What is an inequality?

An inequality is a comparison between two values, numbers, or expressions that aren't equal (but possibly equal).

There are 5 inequality symbols to remember and use:

## How do we solve algebraic inequalities?

We solve inequalities in the exact same way as equations — but equations have one answer, and an inequality has a range of solutions.

When calculating the values of the inequality, we have to pay attention to the direction of the inequality.

## There are rules to solving inequalities!

Let's look at the rules to solve algebraic inequalities using addition, subtraction, multiplication, and division.

x + 9 is greater than or equal to 6.

x + 9 ≥ 6

If we're solving for x by itself, we want to get rid of that 9 next to it, so we subtract 9 from both sides.

x + 9 – 9 ≥ 6 – 9

x is greater than or equal to negative 3.

x ≥ -3

#### Quiz

Express the car speed as an inequality: The speed limit is 30 miles per hour.

## Ready for solving inequalities using subtractions?

### Here's an example:

x minus 4 is less than 8.

x – 4 < 8

First, we're going to add 4 to both sides.

x – 4 + 4 < 8 + 4

Then we simplify.

x < 12

It’s as simple as that!

## Solving inequalities with multiplication and division

There's a special rule for multiplication and division.

When you multiply or divide by a negative number, you have to flip your sign in the opposite direction.

### Let’s look at an example:

Negative 5x is greater than 15.

-5x > 15

To get x by itself, we need to divide both sides by negative 5.

Remember, since we are dividing by -5, we have to flip our inequality sign!

So x is less than negative 3.

x < -3

x over 7 is less than or equal to 4x.

x/7 ≤ 4x

For this inequality, we need to multiply both sides by 7.

### When solving inequalities with multiplication and division, do we flip our sign?

No, we don’t have to since we're multiplying by a positive number.

We’ll multiply both sides by 7, and then we get x, which is less than or equal to 28.

7(x/7) ≤ (7) 4x

x ≤ 28

## Think you've got this? Lets see!

### Solving inequalities: question #1

The number X could be 8 or any number greater than 8.

How could we write this mathematically?

Select one of the following answers:

A: x ≥ 8 B: x = 8 C: x < 8 D: x ≤ 8

### Solving inequalities: question #2

Raymond had 5 candy bars (x). Emily ate some of his candy bars. How many candy bars does Raymond have now?

How could write this mathematically?

Select one of the following answers:

A: x ≥ 5 B: x = 5 C: x < 5 D: x ≤ 5

Why? Inequality sign < indicating less than

### Solving inequalities: question #3

Solve the following inequality for y.

- 5y < 30

Select one of the following answers:

A: y < 6 B: y ≤ 6 C: y ≤ - 6 D: y < - 6

Why? Divide both sides with -5 to get y on its own.

## Combine everything you've learned to far

### Solving inequalities: question #4

Solve the following inequality for y.

2y + 3 ≥ y – 7

Select one of the following answers:

A: 10 ≤ y B: -10 ≤ y C: y ≤ - 5 D: y < 5

### If not....let's give it a try together

Add 7 to both ends of the equation:

2y + 3 + 7 ≥ y – 7 + 7 to get 2y + 10 ≥ y

Now, subtract 2y from each side:

2y - 2y + 10 ≥ y - 2y to get 10y ≥ -y

Divide by negative 1 and flip the sign to get your final answer:

-10 ≤ x

### Let's try an everyday inequality problem:

Peter has \$25 to buy nuts and berries.

• Nuts cost \$5 per kilogram. Peter buys 3 kilograms of nuts

• Berries cost \$7 for a kilogram. How many kilograms does Peter buy?

Which of the following inequalities can be used to find c? Choose your answer:

A. 15 + 7c ≥ 25

B. 15 + 7c ≤ 25

C. 5 + 7c ≥ 25

D. 5 + 7c ≤ 25

#### Quiz

Which of the above inequalities can be used to find c, the possible number of kilograms of berries Peter can buy?

## Take Action

### You can see that solving inequalities in real life is very useful!

Need more exercises or examples?