Welcome to the World of Inequalities!
In your math journey so far, you’ve spent a lot of time finding the exact value of \(x\) (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 something else. Think of a speed limit: you don't have to drive exactly 60 km/h; you just need to keep your speed \(s \le 60\). In this chapter, we will learn how to solve and graph these "ranges" of values for both linear and quadratic expressions.
1. Linear Inequalities
Solving a linear inequality is very similar to solving a normal equation. Your goal is still to get \(x\) all by itself. However, there is one golden rule you must never forget!
The "Negative Flip" Rule
Whenever you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign.
Analogy: Imagine you and a friend are standing on a number line. If you are at 5 and they are at 2, you are "greater." If you both multiply your positions by -1, you are now at -5 and they are at -2. Suddenly, -2 is greater than -5! The relationship flipped because of the negative sign.
Example: Solve \(10 - 2x \le 4\)
- Subtract 10 from both sides: \(-2x \le -6\)
- Divide by -2. Because we are dividing by a negative, flip the sign: \(x \ge 3\)
Quick Review: The Symbols
- \(>\) : Greater than
- \(<\) : Less than
- \(\ge\) : Greater than or equal to
- \(\le\) : Less than or equal to
Key Takeaway: Treat inequalities like equations, but flip the sign if you multiply or divide by a negative number!
2. Quadratic Inequalities
Quadratic inequalities (like \(x^2 - 5x + 6 > 0\)) are a bit sneakier. You cannot just "move numbers around" to solve them. You must follow a specific process involving a sketch.
Step-by-Step Guide to Solving Quadratics
- Rearrange: Make sure one side of the inequality is zero (e.g., \(ax^2 + bx + c > 0\)).
- Find Critical Values: Treat it as an equation (\(= 0\)) and solve it by factorising or using the quadratic formula. These values are where the graph hits the x-axis.
- Sketch the Graph: Draw a quick "u-shape" (if \(x^2\) is positive) passing through your critical values.
- Identify the Region: Look at the original inequality sign:
- If it's \(> 0\), you want the parts of the curve above the x-axis.
- If it's \(< 0\), you want the part of the curve below the x-axis.
Example: Solve \(x^2 - 4x - 5 < 0\)
1. Factorise: \((x - 5)(x + 1) < 0\).
2. Critical values: \(x = 5\) and \(x = -1\).
3. Sketch a "u-shape" crossing at -1 and 5.
4. Since we want \(< 0\), we look below the x-axis. This is the single "valley" between the two points.
Answer: \(-1 < x < 5\)
Don't worry if this seems tricky! Just remember: if the answer is the "valley" below the axis, it's written as one combined inequality. If the answer is the two "arms" pointing up away from the axis, it's written as two separate inequalities (e.g., \(x < -1\) or \(x > 5\)).
Key Takeaway: Always sketch the graph. Never try to solve a quadratic inequality without seeing where the curve goes!
3. Inequalities with Fractions
The Edexcel syllabus mentions that some inequalities involve fractions that can be "reduced" to linear or quadratic ones. A common example is \(\frac{a}{x} < b\).
The Trap!
You might be tempted to just multiply both sides by \(x\). Do not do this! We don't know if \(x\) is positive or negative, so we wouldn't know whether to flip the sign.
The Trick: Multiply by \(x^2\)
Since any number squared (except zero) is always positive, we multiply both sides by \(x^2\) to keep the inequality sign safe.
Example: \(\frac{3}{x} < 2\) becomes \(3x < 2x^2\). Now you have a quadratic inequality that you can solve using the steps in Section 2!
Key Takeaway: If \(x\) is in the denominator, multiply by \(x^2\) to turn it into a quadratic inequality.
4. Graphical Representation and Shading
Sometimes you need to show the solution on a coordinate grid (\(x\) and \(y\)). This is where we represent regions.
Dotted vs. Solid Lines
- Use a dotted line for strict inequalities (\(>\) or \(<\)). This shows the line itself is not part of the solution.
- Use a solid line for "or equal to" inequalities (\(\ge\) or \(\le\)). This shows the line is part of the solution.
How to Shade the Right Area
1. Draw the line: Treat the inequality as an equation (e.g., for \(y > x + 2\), draw the line \(y = x + 2\)).
2. Test a point: Pick a simple coordinate like \((0,0)\) that isn't on the line and plug it into the inequality.
3. Decide: If the point works, shade that side! If it doesn't work, shade the other side.
Did you know? In exams, you are usually asked to shade the region that satisfies the inequalities. However, always read the question carefully—sometimes they ask you to shade the "unwanted" region to leave the correct area white!
Key Takeaway: Dotted = not included. Solid = included. Test \((0,0)\) to find the right side of the line.
Common Mistakes to Avoid
- Forgetting to flip the sign: Always double-check when dividing by a negative number.
- Treating quadratics like linear equations: Never write \(x^2 > 9\) as just \(x > 3\). You are missing the \(x < -3\) part! Sketch it!
- Shading the wrong side: Always use a test point like \((0,0)\) to be 100% sure.
Congratulations! You've covered the core concepts of Inequalities for Pure Math 1. Keep practicing those quadratic sketches—they are the key to mastering this topic!