Chapter 14: Inequalities - More Than Just Equal!

Hey everyone! Ever felt like life isn't always about things being perfectly equal? Sometimes you need at least a certain score to pass a test, or you have at most a certain amount of money to spend. That's where inequalities come in! They are a super useful way to talk about values that are not equal.

In this chapter, we're going to explore the world of inequalities. You'll learn:

• What the different inequality symbols mean.
• How to show inequalities on a number line.
• The special rules for solving inequalities (there's one tricky one to watch out for!).
• How to use inequalities to solve real-life problems.

Don't worry if this sounds new. We'll break it down step-by-step. Let's get started!


What Are Inequalities?

In math, an equation is a statement that two things are equal, using the equals sign ($$=$$). An inequality is a statement that two things are not equal. It tells us about the relationship between them, like if one is bigger, smaller, or maybe bigger-or-equal-to the other.

The Four Symbols You Need to Know

There are four main symbols we use in inequalities:

• > : This means "greater than". For example, $$5 > 2$$ means 5 is greater than 2.

• < : This means "less than". For example, $$3 < 8$$ means 3 is less than 8.

• ≥ : This means "greater than or equal to". For example, $$x ≥ 4$$ means x can be 4, or any number bigger than 4.

• ≤ : This means "less than or equal to". For example, $$y ≤ 10$$ means y can be 10, or any number smaller than 10.

Memory Aid: The Alligator Trick!

A fun way to remember the > and < signs is to think of them as an alligator's mouth. The alligator is always hungry and wants to eat the bigger number!

Example: In $$10 > 1$$, the alligator's mouth is open towards the 10.

Turning Words into Math

We use inequalities to describe situations all the time. Here's how to translate common phrases:

• "is more than" or "is greater than" uses the > sign.
"The number of students (s) is more than 30" becomes $$s > 30$$.

• "is less than" or "is fewer than" uses the < sign.
"The temperature (t) is less than 0" becomes $$t < 0$$.

• "is at least" or "is no less than" uses the ≥ sign.
"You must be at least 12 years old (a) to watch the movie" becomes $$a ≥ 12$$.

• "is at most" or "is no more than" uses the ≤ sign.
"The bag can hold at most 5 kg (w)" becomes $$w ≤ 5$$.

Key Takeaway

Inequalities help us compare quantities that aren't equal. The four symbols are >, <, ≥, and ≤. They can describe many real-world limits and conditions.


Showing Inequalities on a Number Line

A number line is a great way to see all the possible answers to an inequality at once. It gives us a visual picture of the solution.

There are two simple rules for drawing inequalities on a number line.

Rule 1: The Circle

• Use an open circle (○) for > (greater than) and < (less than). This shows that the number itself is not included in the solution.
• Use a closed circle (●) for ≥ (greater than or equal to) and ≤ (less than or equal to). This shows that the number itself is included in the solution.

Easy Tip: If the symbol has a little line underneath it (≥ or ≤), you fill in the circle!

Rule 2: The Arrow

The arrow points in the direction of all the other numbers that are part of the solution.

• For > and ≥, the arrow points to the right (towards the bigger numbers).
• For < and ≤, the arrow points to the left (towards the smaller numbers).

Let's See Some Examples!

1. Represent $$x > 1$$
• We need an open circle at 1 (because it's not "or equal to").
• The arrow points to the right, because we want all numbers greater than 1.


2. Represent $$x ≤ -2$$
• We need a closed circle at -2 (because it is "or equal to").
• The arrow points to the left, because we want all numbers less than or equal to -2.


3. Represent $$x < 4$$
• We need an open circle at 4.
• The arrow points to the left.


4. Represent $$x ≥ 0$$
• We need a closed circle at 0.
• The arrow points to the right.


The Rules for Solving Inequalities

Great news! Solving inequalities is almost exactly like solving equations. You can add, subtract, multiply, and divide to get the variable by itself. However, there is one very important rule you must remember.

The Simple Rules (Just like with equations!)

1. Addition and Subtraction: You can add or subtract the same number from both sides, and the inequality stays the same.
If $$a > b$$, then $$a + c > b + c$$.

2. Multiplication and Division (with POSITIVE numbers): You can multiply or divide both sides by the same positive number, and the inequality stays the same.
If $$a > b$$ and c is positive, then $$ac > bc$$.

THE MOST IMPORTANT RULE! (The Danger Zone)

When you multiply or divide both sides of an inequality by a NEGATIVE number, you MUST FLIP the direction of the inequality sign.

Why? Let's see it with numbers.
We all agree that $$4 > 2$$. This is true.
But what happens if we multiply both sides by -1?
$$4 \times (-1) = -4$$
$$2 \times (-1) = -2$$
Is -4 greater than -2? No! On a number line, -4 is to the left of -2, so it's smaller.
So, we have to flip the sign: $$-4 < -2$$.

Remember: Multiply or divide by a negative? Flip the sign!
• > becomes <
• < becomes >
• ≥ becomes ≤
• ≤ becomes ≥

Key Takeaway

Solving inequalities is like solving equations, but if you multiply or divide by a negative number, you must reverse the inequality symbol. This is the biggest difference and the most common place to make a mistake, so always be careful!


Putting It All Together: Solving Linear Inequalities

Let's practice solving some inequalities step-by-step. Our goal is the same as with equations: get the variable (like 'x') all by itself on one side.

Step-by-Step Guide

1. Use addition or subtraction to move number terms to one side and variable terms to the other.
2. Use multiplication or division to solve for the variable.
3. The Golden Rule Check: Did you just multiply or divide by a negative number? If yes, FLIP the inequality sign!
4. Once you have your solution, you can represent it on a number line.

Example 1: A simple one

Solve $$x + 5 < 12$$

1. We want to get 'x' alone. Let's subtract 5 from both sides.
$$x + 5 - 5 < 12 - 5$$
$$x < 7$$

2. Did we multiply or divide by a negative? No. So the sign stays the same.
3. Answer: $$x < 7$$.
4. On a number line: An open circle at 7, with an arrow pointing to the left.

Example 2: A two-step problem

Solve $$3y - 4 ≥ 11$$

1. First, add 4 to both sides to move the number term.
$$3y - 4 + 4 ≥ 11 + 4$$
$$3y ≥ 15$$

2. Now, divide by 3 to get 'y' alone.
$$\frac{3y}{3} ≥ \frac{15}{3}$$
$$y ≥ 5$$

3. Did we divide by a negative? No, we divided by positive 3. The sign stays the same.
4. Answer: $$y ≥ 5$$.
5. On a number line: A closed circle at 5, with an arrow pointing to the right.

Example 3: The sign FLIP in action!

Solve $$-2x - 1 > 9$$

1. First, add 1 to both sides.
$$-2x - 1 + 1 > 9 + 1$$
$$-2x > 10$$

2. Now, we need to divide by -2 to get 'x' alone. WATCH OUT! We are dividing by a negative number!
$$\frac{-2x}{-2} ? \frac{10}{-2}$$

3. Because we divided by -2, we MUST FLIP the sign. The `>` becomes a `<`.
$$x < -5$$

4. Answer: $$x < -5$$.
5. On a number line: An open circle at -5, with an arrow pointing to the left.


Inequalities in the Real World

The best part about learning this is that you can use it to solve real-life problems!

How to Tackle Word Problems

1. Read and Understand: Figure out what the problem is asking.
2. Define a Variable: Choose a letter (like 'x') to represent the unknown value.
3. Write the Inequality: Translate the words and phrases from the problem into a mathematical inequality.
4. Solve It: Use the rules we just learned to solve the inequality.
5. Answer the Question: Write your answer as a sentence that makes sense in the context of the problem.

Example Problem 1: Concert Tickets

Leo wants to buy concert tickets that cost $40 each. He has a total of $250 to spend. What is the maximum number of tickets he can buy?

1. Variable: Let $$t$$ be the number of tickets Leo can buy.
2. Inequality: The total cost ($$40t$$) must be less than or equal to the money he has ($$250$$).
$$40t ≤ 250$$

3. Solve: Divide both sides by 40 (a positive number, so no sign flip).
$$\frac{40t}{40} ≤ \frac{250}{40}$$
$$t ≤ 6.25$$

4. Answer: Since Leo can't buy a fraction of a ticket, the maximum number he can buy is 6. "The maximum number of tickets Leo can buy is 6."

Example Problem 2: Passing a Course

To pass her Maths course, Maria needs a total score of at least 160 points from two tests. She scored 70 points on her first test. What is the minimum score she needs on her second test?

1. Variable: Let $$s$$ be the score on her second test.
2. Inequality: Her first test score (70) plus her second test score ($$s$$) must be "at least" 160. This means greater than or equal to.
$$70 + s ≥ 160$$

3. Solve: Subtract 70 from both sides.
$$70 - 70 + s ≥ 160 - 70$$
$$s ≥ 90$$

4. Answer: "Maria needs a minimum score of 90 on her second test to pass the course."