Welcome to Algebraic Inequalities!
In most of your maths journey so far, you have dealt with equations—where one side is exactly equal to the other. But in the real world, things aren't always that precise. Sometimes we just need to know if a number is "at least" something or "less than" another.
Think of a speed limit: if the sign says 30mph, you can travel at any speed \( s \) as long as \( s \leq 30 \). In this chapter, you’ll learn how to find, write, and graph these ranges of values. Don't worry if it seems a bit different from normal algebra; the rules are very similar, with just one or two "special twists" to keep an eye on!
1. The Language of Inequalities
Before we solve them, we need to speak the language. There are four main symbols you need to know:
\( < \) : Less than (e.g., \( x < 5 \) means \( x \) could be 4, 3, 2.1... but not 5)
\( > \) : Greater than (e.g., \( x > 10 \) means \( x \) is anything bigger than 10)
\( \leq \) : Less than or equal to (The line underneath means the number itself is included!)
\( \geq \) : Greater than or equal to
Memory Aid: The Hungry Alligator
Imagine the symbol is an alligator's mouth. The alligator is very hungry, so it always opens its mouth to eat the bigger number!
Example: \( 10 > 2 \) (The mouth opens toward the 10).
Quick Review:
- If there is no line underneath, the number is not included.
- If there is a line underneath, the number is included.
2. Representing Inequalities on a Number Line
Sometimes, a picture is worth a thousand words. We use number lines to show which values of \( x \) work for our inequality.
The Two Types of Circles:
1. Open Circle \( \circ \): Used for \( < \) and \( > \). It tells us: "Start here, but don't include this exact number."
2. Closed (Filled) Circle \( \bullet \): Used for \( \leq \) and \( \geq \). It tells us: "This number is part of the answer."
Step-by-Step: Drawing an Inequality
To draw \( x > 3 \):
1. Draw a number line.
2. Put an open circle at 3 (because it's "greater than," not "equal to").
3. Draw an arrow pointing to the right, because the numbers are getting bigger.
Did you know?
If the \( x \) is on the left side (like \( x < 5 \)), the inequality symbol actually points in the direction the arrow should go on the number line! \( < \) points left, so the arrow goes left.
3. Solving Linear Inequalities
Solving an inequality is almost exactly like solving a regular equation (using \( = \)). Your goal is to get the letter (like \( x \)) by itself.
Example: Solve \( 2x + 1 \geq 7 \)
1. Subtract 1 from both sides: \( 2x \geq 6 \)
2. Divide by 2: \( x \geq 3 \)
The answer is \( x \geq 3 \). Simple!
The Golden Rule of Inequalities (Common Mistake Alert!)
There is one situation where inequalities behave differently than equations: When you multiply or divide by a negative number, you MUST flip the inequality sign.
Analogy: Think of a mirror. When you deal with "negative" versions of numbers, everything is reflected and the order of size flips upside down!
Example: Solve \( -3x < 12 \)
Divide by \( -3 \). Because we divided by a negative, we flip \( < \) to \( > \).
\( x > -4 \)
Key Takeaway: Treat the inequality sign like an equals sign, but if you divide or multiply by a minus, flip it!
4. Set Notation (Higher Tier Only)
Sometimes, exam questions will ask you to express your answer in "set notation." This looks fancy, but it's just a formal way of writing your final answer.
Standard answer: \( x \geq 3 \)
Set notation: \( \{x : x \geq 3\} \)
The curly brackets mean "The set of numbers," and the colon \( : \) means "such that." So, you are saying: "The set of all \( x \) values, such that \( x \) is greater than or equal to 3."
5. Quadratic Inequalities (Higher Tier Only)
Quadratic inequalities involve \( x^2 \). They can look scary, but a quick sketch makes them easy to solve.
Step-by-Step Process for \( x^2 - 2x < 3 \):
1. Rearrange to zero: \( x^2 - 2x - 3 < 0 \)
2. Find the "Critical Values": Solve it as if it were an equation \( x^2 - 2x - 3 = 0 \).
Factorising gives \( (x - 3)(x + 1) = 0 \), so \( x = 3 \) and \( x = -1 \).
3. Sketch the graph: Draw a quick U-shaped curve passing through -1 and 3 on the x-axis.
4. Find the region:
- If the question is \( < 0 \), you want the part of the graph below the x-axis (between the two numbers).
- If the question is \( > 0 \), you want the parts above the x-axis (outside the two numbers).
For our example \( x^2 - 2x - 3 < 0 \): The curve is below the axis between -1 and 3.
Answer: \( -1 < x < 3 \)
6. Inequalities in Two Variables (Higher Tier Only)
What happens when we have \( x \) and \( y \)? We represent these as regions on a coordinate graph.
Key Rules for Graphing:
- Dashed Line: Used for \( < \) or \( > \). It means "The boundary is not included."
- Solid Line: Used for \( \leq \) or \( \geq \). It means "The boundary is included."
How to shade the region:
1. Draw the line as if it were an equation (e.g., for \( y > x + 2 \), draw the line \( y = x + 2 \)).
2. Pick a "test point" that isn't on the line. \( (0,0) \) is usually the easiest choice!
3. Plug the test point into the inequality. If it works (e.g., \( 0 > 2 \) is false), then the point is not in the correct region. If it’s false, shade the other side of the line!
Key Takeaway: Always check if your line should be dashed or solid, and use a test point to make sure you are shading the correct side.
Summary Checklist
- Do I know the four symbols and what they mean?
- Can I represent an inequality on a number line using open and closed circles?
- Remembered to flip the sign if multiplying/dividing by a negative?
- (Higher) Can I sketch a quadratic to find the "between" or "outside" regions?
- (Higher) Can I use dashed or solid lines to represent regions on a graph?
Don't worry if this seems tricky at first! Inequalities are just about boundaries. Once you master the "negative flip" rule and the number line circles, you've got this!