# Basic Linear Inequalities

#### Common Questions

#### What is an inequality?

Great question! An example of an inequality is $3＜7$, which we would read as “$3$ is less than $7$.” We can also have variables in inequalities. For example, we could have $x≥4$. We would read this as “$x$ is greater than or equal to $4$.”

Read below to learn more.

#### Common Questions

Great question! An example of an inequality is $3＜7$, which we would read as “$3$ is less than $7$.” We can also have variables in inequalities. For example, we could have $x≥4$. We would read this as “$x$ is greater than or equal to $4$.” Read below to learn more. | |||||||||||

Great question! Here's a chart of some common words associated with each inequality symbol.
Read below to learn more! | |||||||||||

Great question! When solving an inequality, you should reverse the inequality sign whenever you are Read below to learn more. |

### What is an Inequality?

Like an equation, an inequality tells us about the relationship between two values or expressions. However, the two sides of an inequality aren’t necessarily equal. Instead, inequalities tell us if one value is less than or greater than another value using one of the following symbols:

- < (less than)
- > (greater than)
- ≤ (less than or equal to)
- ≥ (greater than or equal to)

Click the button below if you want to review what these symbols mean!

We can write and use inequalities in many real-life situations. Think of an inequality as something that limits how far we can go. For example, there's a limit to how fast you can go on the highway before you're pulled over for speeding and a limit to how long you can procrastinate before your teacher loses it 🥴.

Let's look at some other situations.

### How to Write an Inequality

**Example 1: **A tea shop is selling boba tea at $$2$ per cup and rolled ice cream for $$5$. If you have $$10$ to spend and you must buy yourself one rolled ice cream, how many cups of boba tea can you buy?

First things first, we want to write an inequality for this problem.

Our unknown variable is how many cups of boba tea you can buy, so we'll call that $x$. If each up costs $$2$ and you also have to buy a rolled ice cream for $$5$, which expression best represents the total amount you will spend at the shop?

Great work! The amount of money you have is $$10$, so we need to relate$5+2x$ to $10$ using one of our inequality symbols. Since you can't spend more than $$10$, the amount you spend $5+2x$ must be less than or equal to $10$. That means we'll be using our **less than or equal to **symbol "$≤$".

Putting everything together, we'll get the following inequality:

$5+2x≤10$We can read this inequality as "$5$ plus $2x$ is less than or equal to $10$."

Let's look at another example!

**Example 2: **There are $8$ crewmates and $2$ imposters in a game of Among Us. If the imposters always attack exactly $1$ crewmate every round and the players mistakenly eject $1$ crewmate every round, how many rounds can be played before the number of crewmates is less than or equal to the number of imposters?

Let's call the number of rounds $x$. How many crewmates are lost after $x$ rounds?

Super! One crewmate is attacked and one is ejected every round, so every round you are losing two crewmates. After $x$ rounds, you will have lost $2x$ crewmates total. How many crewmates will be left after $x$ rounds?

Great job! Since you are losing crewmates, you will need to subtract the number of crewmates you have lost from the number of crewmates you had originally.

Which inequality symbol should we use if we want the number of crewmates left, $8−2x$, to be greater than the number of imposters, $2$?

Amazing! The wording of this problem was a bit tricky. We did not want $8−2x$ to equal to $2$, so that means $8−2x$ had to be **more than** $2$. Our inequality is $8−2x>2$ which we can read as "$8$ minus $2x$ is greater than $2$."

Now it's time to try writing a few inequalities on your own!

### Quick Practice 1 / 3

$+$

### How to Solve an Inequality

Solving an inequality will give us a simpler inequality that tells us which values make the original inequality true. We can solve most inequalities exactly like how we would solve an equation.

Let's look at our question from Example 1 again:

A tea shop is selling boba tea at $$2$ per cup and rolled ice cream for $$5$. You have $$10$ to spend, and you must buy yourself some rolled ice cream. The inequality for this problem, $5+2x≤10$, tells you how many cups of boba you can buy, where $x$ is equal to the number of cups of boba.

We want to isolate our variable $x$, so first, let's substract $5$ from both sides:

$5+2x≤10$$2x≤5$Now we can divide both sides of the inequality by $2$:

$2x≤5$$x≤25 $After solving $5+2x≤10$, we get the inequality $x≤25 $. If we were to read it out loud, it would be “$x$ is less than or equal to $25 $.” This means that the number of boba teas you buy must be less than or equal to $25 $. (Keep in mind that while numbers like $25 $ and $−2$ do make the inequality true, you wouldn’t actually be able to buy $25 $ or $−2$ cups of boba tea!).

#### Checking a Solution 🍦

To check if the solution is correct, you can always plug in different values and see if they make the inequality true.

$x$ | $5+2x$ |
---|---|

$−2$ | $1$ |

$0$ | $5$ |

$25 $ | $10$ |

$3$ | $11$ |

$6$ | $17$ |

Which of the following numbers make the inequality $5+2x≤10$ true?

In this inequality, we didn’t need to do anything to our inequality symbol, so solving the inequality was just like solving an equation. However, this won’t always be the case!

### Flipping the Inequality Sign

Think of an inequality like a mirror. Consider the following:

$−1<2$### -2

### -1

### 0

### 1

### 2

We know $−1$ is less than $2$, so this inequality is true. Now what happens if we divide both sides by $−1$?

When we divide $−1$ and $2$ by$−1$, we change their signs to get $1$ and$−2$. Think of it as reflecting $−1$ and $2$ in a mirror.

### -2

### -1

### 0

### 1

### 2

### -2

### -1

### 0

### 1

### 2

### -2

### -1

### 0

### 1

### 2

### -2

### -1

### 0

### 1

### 2

Our new inequality would be

But wait! $1$ is actually bigger than $−2$, so this inequality isn't true anymore! How can we make the inequality true again?

Since we've "reflected" $−1$ and $2$, we also need to reflect our inequality sign. To make the inequality true, we'll flip our less than sign and turn it into a greater than sign.

Our inequality is now:

$1>−2$Now we have "$1$ is greater than $−2$, which is a true statement. When multiplying or dividing an inequality by a negative number, we need to change the direction of our inequality sign.

#### 🙃 Turn that Frown Upside Down!

If you can’t remember when to flip the inequality sign, think of the phrase **“turn that frown upside down.”** Any time you multiply or divide by a **negative (frown)**, you’ll need to **flip** your inequality sign.

Let's look at an example from earlier:

Netflix Premium costs $$18$ a month. If you currently have $$119$ saved up, and you want to have at least $$20$ at all times (in case of emergencies), how many months of Netflix, $x$, will you be able to afford?

Our inequality for this problem is:

$119−18x≥20$If we want to isolate $x$, what should we do first?

Nice! After subtracting $119$ from both sides, we'll have:

$−18x≥−99$What's our next move?

Great job! After dividing both sides by $−18$, we'll have the inequality below. Which inequality symbol should we use?

Awesome work! Our final answer is $x≤5.5$. This tells us that any number less than or equal to $5.5$ should be a solution to our original inequality $119−18x≥20$. (However, since Netflix is a monthly subscription, the maximum number of months you can afford is $5$. You can always try plugging in different numbers to see if this is true!

Now it’s time for you to try solving some inequalities on your own!

### Quick Practice 1 / 3

Nice work! You completed all the questions! 👏🏿👏🏽👏🏻