Welcome to the World of Inequalities!
In most of your math journey so far, you’ve probably spent a lot of time finding exactly what \(x\) is. But in the real world, things aren't always "equal." Sometimes we just need to know if a value is "enough" or "too much." Whether it’s a bridge that can hold at most 20 tonnes or a bank account that needs at least £5 to stay open, we are dealing with Inequalities.
In this chapter, we’ll move beyond the equals sign and learn how to describe ranges of values. Don't worry if this seems tricky at first—once you master the "sketching" trick, you'll find these much easier than they look!
1. Linear Inequalities
Linear inequalities look very similar to linear equations (like \(2x + 3 = 7\)), but they use symbols like \(<\), \(>\), \(\le\), or \(\ge\). You can solve them using the same "balancing" method you use for equations, with one golden rule to remember.
The Golden Rule
When you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign.
Analogy: Think of it like a see-saw. If you are standing on one side and someone "negates" the gravity, everything flips upside down!
Example: Solve \(-3x < 12\).
Divide both sides by \(-3\). Because we divided by a negative, the \(<\) becomes \(>\).
Answer: \(x > -4\).
Solving with Brackets and Fractions
Your MEI syllabus requires you to handle more complex linear inequalities involving brackets and fractions. Just treat them like equations: expand brackets first, and multiply through to clear fractions.
Quick Review:
1. Treat it like an equation.
2. If you multiply/divide by a negative, FLIP THE SIGN.
3. Keep your work tidy to avoid losing track of the symbol!
2. Graphical Representation
Sometimes, a picture is worth a thousand numbers. You need to be able to represent inequalities on a graph, especially when two variables are involved (like \(y > x + 1\)).
How to Draw Inequalities:
- Draw the line: Treat the inequality as an equation (e.g., \(y = x + 1\)).
- Solid or Dashed? Use a dashed line for strict inequalities (\(<\) or \(>\)) because the points on the line are not included. Use a solid line for \(\le\) or \(\ge\).
- Shade the region: Pick a test point (like \((0,0)\)) and see if it makes the inequality true. If it does, shade that side!
Common Mistake: Students often forget to check if the line should be dashed or solid. Memory Aid: If the symbol has a "solid" extra line underneath it (\(\le\)), the line on your graph is "solid" too!
Key Takeaway: Graphical inequalities define a region (an area) rather than just a single point.
3. Quadratic Inequalities
These are a step up in difficulty. A quadratic inequality looks like \(ax^2 + bx + c > 0\). Unlike linear ones, you cannot solve these just by moving numbers around. You must use a sketch.
The 3-Step Method:
- Solve the Equation: Find the "critical values" by setting the quadratic to equal zero. (Factorise it or use the quadratic formula).
- Sketch the Curve: Draw a quick "happy" or "sad" parabola using your critical values as the \(x\)-intercepts.
- Select the Region:
- If the inequality is \(> 0\), you want the part of the curve above the \(x\)-axis.
- If the inequality is \(< 0\), you want the part of the curve below the \(x\)-axis.
Example: Solve \(x^2 - 5x + 6 < 0\).
1. Critical values: \((x - 2)(x - 3) = 0\), so \(x = 2\) and \(x = 3\).
2. Sketch: A "happy" parabola crossing the \(x\)-axis at 2 and 3.
3. Select: Since we want \(< 0\), we look below the axis. This is the "valley" between 2 and 3.
Answer: \(2 < x < 3\).
Did you know? Using a sketch prevents you from making "sign errors." Even top mathematicians use sketches to double-check their logic!
4. Set Notation
In MEI Mathematics B, you are expected to write your answers using Set Notation. This is just a formal way of listing your "allowed" values.
Key Symbols:
- \(\{x : ...\}\) means "The set of \(x\) such that..."
- \(\cup\) (Union) means "OR". Use this for two separate regions (e.g., \(x < 1\) or \(x > 4\)).
- \(\cap\) (Intersection) means "AND". Use this for a single connected region (e.g., \(x > 2\) and \(x < 5\)).
Comparison Table:
Inequality Form: \(2 < x < 5\)
Set Notation Form: \(\{x : x > 2\} \cap \{x : x < 5\}\)
Inequality Form: \(x \le 1\) or \(x \ge 4\)
Set Notation Form: \(\{x : x \le 1\} \cup \{x : x \ge 4\}\)
Summary: Set notation is just a "dressy" way of writing your final answer. Make sure you use the curly brackets!
5. Modulus Inequalities
The modulus sign \(|x|\) means the "magnitude" or "absolute value" of a number. Basically, it’s the distance from zero, ignoring whether it's positive or negative.
Understanding \(|x - a| \le b\)
This is a very common format in MEI exams. Don't let it scare you! It simply means: "The distance between \(x\) and \(a\) is less than or equal to \(b\)."
Analogy: If you are told to stay within 2 meters (\(b\)) of a lamp post (\(a\)), you can move 2 meters to the left or 2 meters to the right.
Example: Solve \(|x - 3| \le 2\).
This means \(x\) is within 2 units of 3.
- 2 units left of 3 is \(3 - 2 = 1\).
- 2 units right of 3 is \(3 + 2 = 5\).
Answer: \(1 \le x \le 5\).
Quick Review:
To solve \(|x - a| \le b\), just write it as: \(a - b \le x \le a + b\). It’s that simple!
Final Summary: Key Takeaways
- Linear: Flip the sign if you multiply/divide by a negative.
- Graphical: Solid lines for \(\le\) and \(\ge\); dashed lines for \(<\) and \(>\).
- Quadratic: Always, always SKETCH the parabola to find your regions.
- Modulus: Think of it as "distance from a point."
- Set Notation: Use \(\cup\) for outside regions (OR) and \(\cap\) for inside regions (AND).
Keep practicing! Inequalities are all about visualizing the number line. Once you see the "gaps" and "regions" in your head, you'll be an expert.