Welcome to the World of Inequalities!
In your previous math classes, you spent a lot of time finding the exact value of \(x\) (like \(x = 5\)). In this chapter, we are looking at Inequalities. Instead of just one answer, we are looking for a range of possible values.
Think of it like a speed limit: if the sign says 70 mph, you don't have to drive exactly 70. You can drive any speed \(s\) as long as \(s \le 70\). Inequalities help us describe these real-world limits, from bridge weight capacities to the temperature required for a chemical reaction to occur. Don't worry if this seems a bit different at first—by the end of these notes, you'll be a pro at finding and shading these regions!
1. Linear Inequalities in One Variable
Solving a linear inequality is almost exactly like solving a linear equation, but with one very important "Golden Rule."
The Basics
We use four main symbols:
\( > \) : Greater than
\( < \) : Less than (Tip: The symbol looks like a squashed letter "L" for Less than!)
\( \ge \) : Greater than or equal to
\( \le \) : Less than or equal to
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.
Example: If we have \( -2x < 10 \), we divide by \( -2 \). The sign flips:
\( x > -5 \)
Why? Think of a number line. \( 2 < 5 \) is true. But if we multiply by \( -1 \), is \( -2 < -5 \)? No! \( -2 \) is actually "warmer" (greater) than \( -5 \). So we must flip it to \( -2 > -5 \) to keep the statement true.
Step-by-Step: Solving with Brackets and Fractions
1. Expand any brackets.
2. Clear fractions by multiplying every term by the denominator.
3. Rearrange to get all \(x\) terms on one side and numbers on the other.
4. Solve for \(x\) (remembering to flip the sign if you divide by a negative!).
Quick Review: Treating an inequality like an equation works 99% of the time. Just watch out for those negative numbers!
2. Representing Inequalities Graphically
Sometimes, we need to show an inequality on a coordinate grid (an \(x-y\) graph). This is common for Ref Ma7, such as \( y > x + 1 \).
The Boundary Line
First, pretend the inequality is an equals sign and draw the line (e.g., \( y = x + 1 \)).
- Use a Dashed/Dotted Line for \( < \) or \( > \). This shows the boundary itself is not included.
- Use a Solid Line for \( \le \) or \( \ge \). This shows the boundary is included.
Shading the Region
To decide which side of the line to shade, use the "Test Point" Method:
1. Pick a point not on the line (the origin \( (0,0) \) is usually the easiest!).
2. Plug the \(x\) and \(y\) values into your inequality.
3. If the statement is true, shade the side containing that point. If it's false, shade the other side.
Did you know? In MEI exams, you must state clearly which region you are identifying. Some people shade the "wanted" region, and some shade the "unwanted" region. Always read the question to see if it asks you to "shade the region that satisfies the inequality."
3. Quadratic Inequalities
Solving quadratic inequalities (Ref a8) like \( x^2 - 5x + 6 < 0 \) requires a slightly different approach. You cannot just "solve for \(x\)" directly.
The 3-Step Method
Step 1: Find the "Critical Values"
Treat the inequality as an equation and solve it (factorising is usually best).
\( x^2 - 5x + 6 = 0 \)
\( (x - 2)(x - 3) = 0 \)
Critical values: \( x = 2 \) and \( x = 3 \).
Step 2: Sketch the Curve
Draw a quick sketch of the parabola. Since \( x^2 \) is positive, it’s a "happy" U-shaped curve crossing the \(x\)-axis at 2 and 3.
Step 3: Identify the Region
Look at the original inequality sign:
- If it is \( < 0 \), you want the part of the curve below the \(x\)-axis. This is usually one single interval: \( 2 < x < 3 \).
- If it is \( > 0 \), you want the parts above the \(x\)-axis. These are two separate tails: \( x < 2 \) or \( x > 3 \).
Common Mistake: Many students see \( x^2 > 9 \) and just write \( x > 3 \). They forget the negative side! If you sketch the curve, you'll see the answer is actually \( x > 3 \) or \( x < -3 \).
4. Set Notation and "And/Or"
Ref a9 requires you to write your answers using proper notation. There are three common ways to write the same thing:
1. Simple Inequalities
Example: \( x < 1 \) or \( x > 4 \)
2. Using 'And' vs 'Or'
- 'Or' (Union \( \cup \)): Used when the regions are separate (like the two tails of a quadratic).
- 'And' (Intersection \( \cap \)): Used when the value must satisfy both conditions at once (like the middle section of a quadratic).
3. Set Notation
This looks a bit fancy but is easy once you break it down:
\( \{x : x > 4\} \)
This is read as: "The set of all \(x\), such that \(x\) is greater than 4."
- The curly brackets \( \{ \} \) mean "The set of..."
- The colon \( : \) means "such that..."
Key Takeaway:
Separate regions? Use or / \( \cup \).
Between two numbers? Use and / \( \cap \).
Summary Checklist for Success
- Did I flip the sign if I divided by a negative?
- For quadratic inequalities, did I draw a sketch? (Always sketch!)
- Is my boundary line dashed (strictly greater/less) or solid (equal to)?
- Have I checked if the question asks for "Set Notation"?
- Did I test a point (like \(0,0\)) to make sure I shaded the correct side?
Don't worry if the set notation looks like a foreign language at first. Just remember the colon means "such that," and the rest is just the inequality you already solved!