Welcome to Linear and Quadratic Inequalities
In your previous math studies, you’ve spent a lot of time solving equations to find the exact value of an unknown (like \(x = 5\)). But in the real world, things aren't always that precise! Sometimes we just need to know if a value is "greater than" or "less than" a certain limit. This is where inequalities come in.
In this chapter, we will master how to solve linear and quadratic inequalities and how to represent them on a graph. These skills are essential for the Linear Programming problems you might face in your exams.
1. Solving Linear Inequalities
Linear inequalities look very similar to linear equations, such as \(ax + b > cx + d\). The good news is that you solve them almost exactly the same way as equations, with one very important exception.
The Golden Rule of Inequalities
When you multiply or divide both sides of an inequality by a negative number, you MUST flip the inequality sign.
For example, if \(-2x < 10\), and you divide by \(-2\), the sign flips: \(x > -5\).
Step-by-Step Example
Solve the inequality: \(3x + 7 > 5x - 3\)
1. Move all the \(x\) terms to one side: \(7 > 2x - 3\) (by subtracting \(3x\) from both sides).
2. Move the constants to the other side: \(10 > 2x\) (by adding \(3\) to both sides).
3. Divide by 2: \(5 > x\), which is the same as \(x < 5\).
Quick Review: Remember the symbols!
\(>\) : Greater than (open circle/dotted line)
\(<\) : Less than (open circle/dotted line)
\(\geq\) : Greater than or equal to (solid circle/solid line)
\(\leq\) : Less than or equal to (solid circle/solid line)
2. Solving Quadratic Inequalities
Quadratic inequalities involve an \(x^2\) term, such as \(px^2 + qx + r < 0\). These are a bit more complex because the solution isn't just a single direction; it's often a range or two separate regions.
The Three-Step Method
Step 1: Find the "Critical Values"
Treat the inequality as an equation and solve \(px^2 + qx + r = 0\). You can do this by factorising, using the quadratic formula, or completing the square (concepts you learned in Section 2 of this course). The solutions are your critical values.
Step 2: Sketch the Graph
Sketch a quick parabola (\(y = ax^2 + bx + c\)). Since \(a\) is usually positive in these questions, it will be a "U-shaped" curve. Mark your critical values on the \(x\)-axis.
Step 3: Identify the Region
Look at the original inequality sign:
- If it is \( < 0 \) or \( \leq 0 \), you want the part of the curve below the \(x\)-axis (the "valley" between the roots).
- If it is \( > 0 \) or \( \geq 0 \), you want the parts of the curve above the \(x\)-axis (the "tails" outside the roots).
Example: \(x^2 - 5x + 6 < 0\)
1. Solve \(x^2 - 5x + 6 = 0 \implies (x - 2)(x - 3) = 0\). Critical values are \(x = 2\) and \(x = 3\).
2. Sketch a U-shaped curve crossing the \(x\)-axis at 2 and 3.
3. Since we want \(< 0\), we look below the axis. The solution is the single region between the roots: \(2 < x < 3\).
Key Takeaway: If the quadratic is \(> 0\), your answer will usually look like two separate inequalities (e.g., \(x < 2\) or \(x > 3\)). If it is \(< 0\), it will be one combined inequality (e.g., \(2 < x < 3\)).
3. Graphical Representation of Linear Inequalities
When we have two variables (usually \(x\) and \(y\)), we represent inequalities by shading regions on a coordinate plane.
How to Draw the Boundary
1. Treat the inequality as an equation (e.g., change \(y \geq 2x + 1\) to \(y = 2x + 1\)).
2. Draw the line. Use a solid line for \(\leq\) or \(\geq\) and a dotted line for \(<\) or \(>\).
3. Shading: To find out which side of the line to shade, pick a test point not on the line (the origin \((0,0)\) is usually the easiest). Plug it into the inequality. If it makes the inequality true, shade that side! If it's false, shade the other side.
Common Mistake: Don't assume \(>\) always means "shade above." Always use a test point to be 100% sure!
4. Simple Linear Programming
Linear programming is a fancy name for finding the "best" solution given a set of constraints (inequalities). In Further Pure Maths, you will typically be asked to identify a Feasible Region.
Finding the Feasible Region
The feasible region is the area on the graph where all the given inequalities are satisfied at once. It is the "overlap" of all your shaded areas.
Example scenario:
Suppose you are given:
\(y \leq 4\)
\(x \geq 1\)
\(y \geq x\)
You would draw the horizontal line \(y = 4\), the vertical line \(x = 1\), and the diagonal line \(y = x\). The triangle where all three conditions meet is your feasible region. In the exam, you are often asked to label this region with a specific letter, like R.
Did you know? Linear programming is used by airlines to decide flight paths and by factories to minimize waste. It’s all about finding the "sweet spot" within limits!
Chapter Summary
Linear Inequalities: Solve like equations, but flip the sign if you multiply/divide by a negative number.
Quadratic Inequalities: Find critical values, sketch the parabola, and choose the region (inside the roots for \(<\), outside for \(>\)).
Graphs: Solid lines for inclusive inequalities (\(\leq, \geq\)), dotted lines for strict ones (\(<, >\)).
Feasible Region: The shared area on a graph that satisfies multiple inequalities at once.
Note: For more complex equations involving cubics or simultaneous linear/quadratic equations, please refer to the other chapters in the "Identities and Inequalities" section.