Maths (0580) Study Notes: The Chapter on Inequalities
Welcome to the exciting world of Inequalities! You’ve spent a lot of time solving equations (where the answer is exactly equal to one number). But in real life, things are rarely exact. You might need at least 5 litres of petrol, or the maximum weight allowed might be less than 50 kg.
Inequalities deal with these kinds of limits. They describe a range of values that a variable can take. This chapter links closely with Algebra and Graphs, helping you visualize these ranges. Don't worry if this seems tricky at first; we will break down the rules step-by-step!
1. Understanding Inequality Symbols
The core of inequalities lies in understanding the four main symbols:
- \(<\) : Less than (Strict inequality)
- \(>\) : Greater than (Strict inequality)
- \(\le\) : Less than or equal to (Inclusive inequality)
- \(\ge\) : Greater than or equal to (Inclusive inequality)
Quick Review Box: The Symbols
- Think of it like a hungry crocodile: The open side of the symbol always faces the larger quantity or expression.
- If the symbol has a line underneath it (\(\le\) or \(\ge\)), it means the end value is included in the solution.
Analogy: The Speed Limit
If the sign says the speed limit is 40 mph, the mathematical inequality for your speed (\(s\)) is \(s \le 40\). You can drive 40 mph or anything less. But if the sign says you must be driving faster than 5 mph to use the express lane, that's a strict inequality: \(s > 5\).
2. Representing Inequalities on a Number Line (C2.6 / E2.6.1)
A number line is the best way to visualize all the numbers that satisfy an inequality. We use two different types of circles at the boundary point:
2.1 Open Circles (Strict Inequalities)
If the inequality is strict (\(<\) or \(>\)), we use an open circle (\(\circ\)) to show that the number itself is NOT included in the solution.
- Example 1: \(x > -2\)
This means \(x\) can be -1.9, -1.999, but *not* exactly -2.
Visualization: An open circle at -2, and an arrow pointing right (towards greater numbers).
2.2 Closed Circles (Inclusive Inequalities)
If the inequality is inclusive (\(\le\) or \(\ge\)), we use a closed (or solid) circle (\(\bullet\)) to show that the number IS included in the solution.
- Example 2: \(x \le 5\)
This means \(x\) can be exactly 5, or any number less than 5.
Visualization: A closed circle at 5, and an arrow pointing left (towards smaller numbers).
Key Takeaway for Number Lines: Line underneath the symbol (\(\le, \ge\)) means dot is solid (\(\bullet\)). No line underneath (\(<, >\)) means dot is open (\(\circ\)).
3. Solving Linear Inequalities in One Variable (C2.6 / E2.6.2)
Solving a linear inequality is almost exactly the same as solving a linear equation. You can add, subtract, multiply, and divide terms on both sides to isolate the variable (\(x\)).
3.1 The Golden Rule: Flipping the Sign
This is the most important difference between equations and inequalities, and it’s where students often make mistakes!
You must FLIP the direction of the inequality sign whenever you multiply or divide both sides by a negative number.
Example of the Golden Rule:
Start with: \(10 > 5\). This is true.
Now, divide both sides by \(-5\):
Left side: \(10 \div (-5) = -2\)
Right side: \(5 \div (-5) = -1\)
Since \(-2\) is less than \(-1\), we must flip the sign:
New inequality: \(-2 < -1\). (True)
3.2 Step-by-Step Solving
Example 3: Solve \(3x + 4 \le 16\)
- Subtract 4 from both sides:
\(3x \le 16 - 4\)
\(3x \le 12\) - Divide by 3 (a positive number, so the sign stays the same):
\(x \le 4\)
Interpretation: The solution includes 4 and all numbers smaller than 4.
Example 4: Solve \(5 - 2x > 1\)
- Subtract 5 from both sides:
\(-2x > 1 - 5\)
\(-2x > -4\) - Divide by -2 (a negative number, so FLIP the sign):
\(x < (-4) \div (-2)\)
\(x < 2\)
Common Mistake to Avoid: Don't flip the sign just because there is a negative number in the question (like the \(-4\) in step 1). Only flip if the number you are *multiplying or dividing by* is negative.
Key Takeaway: Solve like an equation, but be vigilant about the golden rule! If you multiply/divide by a negative, reverse the inequality sign.
4. Compound Inequalities (C2.6 / E2.6.2)
Sometimes a variable has limits on both sides, creating a compound inequality. These look like two inequalities joined together, often using the word "and" or "or." The syllabus focuses on the "and" type.
Example 5: Solve \(-3 < 3x - 2 < 7\)
To solve this, we want to isolate \(x\) in the middle. Whatever operation we perform, we must do it to all three parts of the inequality.
- Add 2 to all three parts: (This removes the -2 from the middle)
\(-3 + 2 < 3x - 2 + 2 < 7 + 2\)
\(-1 < 3x < 9\) - Divide all three parts by 3: (3 is positive, so signs stay the same)
\(-1 \div 3 < x < 9 \div 3\)
\(-0.33... < x < 3\)
Interpretation: \(x\) is any value strictly between \(-1/3\) and 3.
Representing on a Number Line:
You would draw an open circle at \(-1/3\) and an open circle at 3, and then shade the line segment connecting them. The solution is the set of values *between* the two dots.
Key Takeaway: Treat the left, middle, and right sides equally. Perform the same operation on all three parts until \(x\) is alone in the middle.
(Extended Content Only)
5. Graphing Linear Inequalities in Two Variables (E2.6.4, E2.6.5)
When you have inequalities involving both \(x\) and \(y\) (like \(y < 2x + 1\)), the solution is no longer a line on a number line, but an entire region on a Cartesian graph.
Solving these involves drawing a boundary line and then shading the required region.
Step 1: Determine the Boundary Line and its Style
First, ignore the inequality sign and draw the equation of the line (\(y = mx + c\)).
-
Strict Inequalities (< or >): Use a broken (or dashed) line.
Why? Because points *on* the line are not included in the solution. -
Inclusive Inequalities (\(\le\) or \(\ge\)): Use a solid line.
Why? Because points *on* the line are included in the solution.
Step 2: Determine the Region to Shade (Unwanted Region)
The solution to an inequality is one side of the boundary line. To decide which side to shade, we use a test point.
The easiest test point is usually the origin, \((0, 0)\), unless the line passes through it.
- Pick a test point (e.g., \((0, 0)\)).
- Substitute the coordinates into the inequality.
- If the statement is TRUE, that region is the WANTED region.
- If the statement is FALSE, that region is the UNWANTED region.
Critical Syllabus Instruction: In Cambridge IGCSE, unless otherwise directed, you should shade the UNWANTED region. This leaves the solution region clear and unshaded.
Example 6: Graph the inequality \(y < 2x + 1\)
- Boundary Line: \(y = 2x + 1\). Since it's \(<\), use a broken line.
- Test Point: Use \((0, 0)\).
- Substitution: \(0 < 2(0) + 1 \implies 0 < 1\). This is TRUE.
- Shading: Since \((0, 0)\) is in the TRUE (Wanted) region, we shade the opposite side (the Unwanted region).
5.3 Defining Regions with Multiple Inequalities (E2.6.5)
When solving systems of inequalities, you will draw several boundary lines, and the solution will be the region that satisfies *all* the inequalities simultaneously.
Example 7: List the inequalities that define the unshaded region R, bounded by lines L1, L2, and L3.
Let's assume:
- L1 is the horizontal line \(y = 2\) (Solid line)
- L2 is the vertical line \(x = -1\) (Broken line)
- L3 is the diagonal line \(y = x + 4\) (Solid line)
If Region R is below L1, to the right of L2, and above L3, the inequalities defining R are:
-
For L1 (\(y = 2\)): R is below the solid line.
Inequality: \(y \le 2\) -
For L2 (\(x = -1\)): R is to the right of the broken line.
Inequality: \(x > -1\) -
For L3 (\(y = x + 4\)): R is above the solid line. (Using test point (0, 0) gives \(0 > 4\), which is FALSE. So, R is the 'True' region).
Inequality: \(y \ge x + 4\)
The region R is defined by the three inequalities: \(y \le 2\), \(x > -1\), and \(y \ge x + 4\).
Did You Know? Inequalities are used heavily in logistics and business planning (called "Linear Programming") to find the best way to allocate resources or maximize profit, based on various limits and constraints!
Key Takeaway for Graphing: Use solid lines for \(\le, \ge\) and broken lines for \(<, >\). Always use a test point to find the true region, and then typically shade the UNWANTED region to leave the solution clear.