Welcome to the World of Inequalities!
In this chapter, we are moving beyond simple equations where things are "equal" and exploring the world of inequalities—where things are greater than, less than, or somewhere in between. Think of it like a speed limit: you don't have to drive exactly 60 mph; you just have to stay at or below it. Inequalities allow us to describe ranges and regions, which is essential for everything from engineering tolerances to financial forecasting.
Don’t worry if this seems a bit abstract at first. By the end of these notes, you'll be able to solve linear and quadratic inequalities with confidence and even draw them on a graph!
1. The Basics: Linear Inequalities
Solving a linear inequality is very similar to solving a normal linear equation (like \(2x + 3 = 7\)). Your goal is still to get \(x\) all by itself.
The Golden Rule
There is one massive difference you must remember: If you multiply or divide an inequality by a negative number, you must flip the inequality sign.
Example: If you have \(-2x < 10\), and you divide by \(-2\), the answer becomes \(x > -5\).
Step-by-Step Example
Solve \(3(x - 2) \leq 12 + x\):
1. Expand the brackets: \(3x - 6 \leq 12 + x\)
2. Move all \(x\) terms to one side: \(2x - 6 \leq 12\)
3. Move the numbers to the other side: \(2x \leq 18\)
4. Divide by 2: \(x \leq 9\)
Quick Review Box:
\( > \) : Greater than
\( < \) : Less than
\( \geq \) : Greater than or equal to
\( \leq \) : Less than or equal to
Key Takeaway: Treat inequalities like equations, but always flip the sign if you multiply or divide by a negative!
2. Expressing Your Answers
In A Level Maths, the way you write your answer is just as important as the answer itself. You need to be comfortable with three styles:
A. Set Notation
This looks a bit fancy but it’s quite simple. We use curly brackets \(\{ \}\) to describe a "set" of numbers.
Example: \(\{x : x > 3\}\) means "the set of all values of \(x\) such that \(x\) is greater than 3."
B. Interval Notation
This is a shorthand way of writing ranges:
- Use square brackets \([ \,\, ]\) if the number is included (\(\leq\) or \(\geq\)).
- Use round brackets \(( \,\, )\) if the number is not included (\(<\) or \(>\)).
Example: \((2, 5]\) means \(2 < x \leq 5\).
C. 'And' vs 'Or'
- And (\(\cap\)): Used when \(x\) must satisfy two conditions at the same time (the overlap).
- Or (\(\cup\)): Used when \(x\) can be in one region or another (the union).
Memory Aid: Think of Union as a bucket that collects everything from both sides!
Common Mistake: Writing \(5 < x < 2\). This is impossible! A number cannot be bigger than 5 and smaller than 2 at the same time. You should write \(x > 5\) or \(x < 2\).
3. Quadratic Inequalities
Quadratic inequalities (like \(x^2 - 5x + 6 < 0\)) are a bit more complex. You cannot just "solve for \(x\)" directly. You must follow these steps:
The Three-Step Method
Step 1: Find the Critical Values.
Pretend it’s an equation (\(= 0\)) and solve it (usually by factorising).
Example: \(x^2 - 5x + 6 = 0 \Rightarrow (x - 2)(x - 3) = 0\). The critical values are \(x = 2\) and \(x = 3\).
Step 2: Sketch the Graph.
Draw a quick sketch of the parabola. Since the \(x^2\) is positive, it’s a "happy" U-shape crossing the \(x\)-axis at 2 and 3.
Step 3: Identify the Region.
- If the inequality is \( < 0 \), you want the part of the curve below the \(x\)-axis.
- If the inequality is \( > 0 \), you want the parts above the \(x\)-axis.
In our example (\( < 0 \)): The curve is below the axis between 2 and 3, so \(2 < x < 3\).
Key Takeaway: Never try to solve a quadratic inequality without a sketch! The sketch tells you if the answer is one single range or two separate parts.
4. Graphical Inequalities
Sometimes you need to represent inequalities on a coordinate grid (\(x\) and \(y\)). This is great for visualizing "allowed regions."
Rules for Shading:
1. The Line: Draw the line/curve for the equation.
- Use a solid line for \(\leq\) or \(\geq\).
- Use a dashed line for \(<\) or \(>\).
2. The Region:
- \(y > f(x)\) means the region above the line.
- \(y < f(x)\) means the region below the line.
Did you know? This technique is the basis of "Linear Programming," which companies like Amazon use to calculate the most efficient delivery routes!
5. Modulus Inequalities (Stage 2)
The modulus \(|x|\) simply means the positive version of a number. It represents the distance from zero.
Example: \(|x - 3| < 5\).
This means the distance between \(x\) and 3 is less than 5 units.
To solve this, you can rewrite it as: \(-5 < x - 3 < 5\).
Add 3 to everything: \(-2 < x < 8\).
Quick Trick: For \(|f(x)| < a\), write \(-a < f(x) < a\). For \(|f(x)| > a\), write \(f(x) > a\) or \(f(x) < -a\).
Summary Takeaway: Modulus inequalities are just two inequalities hiding in one! Break them out and solve them normally.
Final Quick Review
1. Linear: Isolate \(x\). Flip the sign for negative multiplication/division.
2. Quadratic: Factorise to find critical values, sketch the graph, then pick the region (inside the U for \(<\), outside for \(>\)).
3. Notation: Use \([ \, ]\) for "including" and \(( \, )\) for "excluding."
4. Graphs: Dashed lines for strict inequalities (\(<\), \(>\)), solid for "or equal to" (\(\leq\), \(\geq\)).
5. Modulus: Think of distance and split the inequality into two parts.
Don't worry if this feels like a lot to take in! Start with the linear ones, and once you're comfortable sketching parabolas, the quadratic inequalities will fall into place. You've got this!