Understanding Inqualities: The Maths of Comparing numbers

Inequalities vs. Equations

You’ve probably already met equations, like “1 + 1 = 2”, which say two sides are perfectly equal. Inequalities, on the other hand, deal with situations where things aren’t equal. They use symbols such as “greater than” (>) and “less than” (<) to show relationships. For instance, “5 > 3” tells us that 5 is greater than 3. Imagine these symbols as hungry alligator mouths that always want to gobble up the bigger number. The open end of the symbol faces the larger value because that’s what the alligator wants to “eat.”

But here’s something important: the order of numbers matters! For example, “3 > 5” doesn’t work because the smaller number (3) is incorrectly paired with the greater-than symbol. To fix it, you flip the symbol so it reads “3 < 5”. This simple rule helps you avoid confusion when writing inequalities.

A Visual Tool: The Number Line

A number line is like a maths playground for inequalities. It stretches endlessly in both directions, with numbers increasing as you move to the right and decreasing as you move left. Inequalities come alive on the number line because they help us represent whole sets of solutions visually. For instance, if we’re asked to graph “n > 3”, we start with a hollow circle at 3 (to show 3 isn’t included) and then draw a line or arrow to the right to indicate all numbers greater than 3.

Now, if the inequality changes to “n ≥ 3”, we fill in the circle at 3 because this time the number 3 is included in the solution set. This simple difference between a hollow and solid circle makes it clear which numbers satisfy the inequality.

Why Do We Use Inequalities?

Inequalities are not just maths tricks—they’re incredibly useful in everyday life. Think about when you go to an amusement park, and a ride says you must be “10 or older” to ride. That’s an inequality written as “Age ≥ 10”. The filled-in circle at 10 includes your age, so you can ride. But if the rule said “older than 10”, represented as “Age > 10”, you’d be left out because 10 isn’t greater than 10.

Another example could be when you’re shopping for a new bike. If your budget is between £50 and £200, you’d use the inequality “50 ≤ P ≤ 200”, where “P” represents the price. Here, both £50 and £200 are included in your budget, which would be shown with filled-in circles on a number line.

The Symbols: Greater Than, Less Than, and Beyond

The basic inequality symbols “>” (greater than) and “<” (less than) are the building blocks of comparison. But did you know there are also combined symbols? The “≥” symbol means “greater than or equal to”, and “≤” means “less than or equal to”. These symbols are perfect for situations where the boundary number matters, like the amusement park example where age 10 is included in “Age ≥ 10”.

Graphing Inequalities: A Closer Look

Let’s get back to the number line. Why do we use hollow and solid circles? Imagine you’re graphing “n > 3”. The hollow circle at 3 shows that 3 isn’t part of the solution, while the arrow to the right shows all numbers greater than 3 are included. If we change it to “n ≥ 3”, the solid circle at 3 now says, “Yes, 3 counts too!”

But inequalities can get even cooler. What if we want to find numbers between 3 and 7? This is called a compound inequality, written as “3 < n < 7”. On a number line, you’d draw a line from 3 to 7 with hollow circles at both ends, showing the range of numbers that work.

Real-Life Applications

Inequalities are everywhere in real life. Besides amusement parks and shopping, they’re used to describe temperature ranges, speed limits, or even exam results. For example, if your cat prefers the thermostat to be set between 68 and 72 degrees Fahrenheit, you’d write “68 ≤ T ≤ 72”. The filled-in circles mean your cat would be happy if it’s exactly 68 or 72 degrees, too.

Why It Matters

Learning inequalities isn’t just about passing your maths exam—it’s about understanding how to describe limits, boundaries, and ranges in everyday life. Whether you’re solving puzzles, setting goals, or comparing options, inequalities help you make sense of the world. So next time you see “0.6 < 1.5”, just remember: it’s the alligator’s way of telling you which number wins!