### 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**.

### Let's start with algebraic inequalities using addition.

**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

This gives us our answer.

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.

If you are multiplying or dividing by a **positive number**, **don’t worry** about this step.

### 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

The correct answer is: **A**

Why? Inequality sign **≥** indicating **greater or equal to**

### 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

The correct answer is: **C**

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

The correct answer is: **D**

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

The correct answer is: **B**

### Did you get that last answer?

### 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?

### Try these links.

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