Welcome to the World of Inequalities!
In your previous math studies, you spent a lot of time finding the exact value of \(x\). But in the real world, things aren't always so precise. Think about a speed limit: you don't have to drive at exactly 30 mph; you just need to stay below or equal to 30. That is an inequality!
In this chapter, we are going to learn how to solve and graph these "ranges" of values. Don't worry if this seems tricky at first—once you learn a few simple rules and how to draw a quick sketch, you'll find that inequalities are just as logical as equations.
1. Solving Linear Inequalities
Linear inequalities look like linear equations (e.g., \(3x + 1 < 10\)). You solve them using the same steps you use for equations: adding, subtracting, multiplying, and dividing to get \(x\) on its own.
The Golden Rule of Inequalities
There is one vital difference between equations and inequalities:
If you multiply or divide by a negative number, you MUST flip the inequality sign!
Example: If you have \( -2x < 10 \), and you divide by \( -2 \), the sign flips: \( x > -5 \).
Analogy: Think of the inequality sign like a weather vane. Usually, it points one way, but if a "negative" wind blows (multiplication or division), it flips to face the opposite direction!
Solving "Double" Inequalities
Sometimes you’ll see an \(x\) sandwiched between two signs, like this: \(10 < 3x + 1 < 16\).
The goal is to get \(x\) alone in the middle. Whatever you do to the middle, you must do to both the left and the right sides.
Step-by-Step Example: Solve \(10 < 3x + 1 < 16\)
1. Subtract 1 from all three parts: \(9 < 3x < 15\)
2. Divide all three parts by 3: \(3 < x < 5\)
This means \(x\) can be any number between 3 and 5, but not 3 or 5 themselves.
Quick Review:
• \( > \) means "Greater than"
• \( < \) means "Less than"
• \( \ge \) means "Greater than or equal to"
• \( \le \) means "Less than or equal to"
Key Takeaway: Treat linear inequalities like equations, but always remember to flip the sign if you multiply or divide by a negative!
2. Quadratic Inequalities
Quadratic inequalities involve an \(x^2\) term, like \((2x+5)(x+3) > 0\). You cannot solve these by just moving numbers around. You must follow a specific process.
The 4-Step Method
1. Find the Critical Values: Treat the inequality as an equation (set it to 0) and solve it to find the roots. For \((2x+5)(x+3) = 0\), the roots are \(x = -2.5\) and \(x = -3\). These are our "boundary lines."
2. Sketch the Graph: Draw a quick sketch of the parabola. Since the \(x^2\) term is positive (if we expanded it), it’s a "happy" U-shape curve. Mark your critical values on the x-axis.
3. Look at the Sign:
• If the inequality is \( > 0 \), you want the parts of the curve above the x-axis.
• If the inequality is \( < 0 \), you want the part of the curve below the x-axis.
4. Write the Final Solution:
For \((2x+5)(x+3) > 0\), we want the parts above the axis. This happens in two separate "tails": \(x < -3\) or \(x > -2.5\).
Did you know?
A common mistake is trying to solve quadratic inequalities using logic alone without a sketch. Always sketch the curve! It only takes 5 seconds and prevents 90% of errors.
Key Takeaway: Use the critical values to find where the graph hits the x-axis, then use your sketch to decide if you need the "middle bit" (below the axis) or the "outside tails" (above the axis).
3. Expressing Your Answer (Notation)
Mathematicians have specific ways of writing down ranges of numbers. You need to be comfortable with Set Notation and Interval Notation.
Set Notation
This looks a bit scary but it's just a formal "wrapper" for your answer.
Example: \( \{x : x > 3\} \)
This is read as: "The set of all \(x\) such that \(x\) is greater than 3."
If you have two separate regions, use these symbols:
\(\cup\) (Union): Used for "OR". Example: \( \{x : x < -3\} \cup \{x : x > -2.5\} \)
\(\cap\) (Intersection): Used for "AND" (where regions overlap).
Interval Notation
This is a shorthand way to write ranges using brackets.
• Round brackets \(( )\) mean the end number is not included (for \( < \) or \( > \)).
• Square brackets \([ ]\) mean the end number is included (for \( \le \) or \( \ge \)).
Examples:
• \(2 < x < 3\) becomes \( (2, 3) \)
• \(2 \le x < 3\) becomes \( [2, 3) \)
• \(x \ge 2\) becomes \( [2, \infty) \) (Infinity always gets a round bracket because you can't "reach" it!)
Memory Trick:
Square brackets look like a box—they "hold onto" the number. Round brackets are like open hands—the number slips through!
Key Takeaway: Master the brackets! \(( \text{not included} )\) and \([ \text{included} ]\).
4. Graphical Representation of Inequalities
You can also show inequalities on a 2D coordinate grid with \(x\) and \(y\).
Linear Graphs (e.g., \(y > x + 1\))
1. Draw the line \(y = x + 1\).
2. Dashed or Solid? If it's \( > \) or \( < \), use a dashed line. If it's \( \ge \) or \( \le \), use a solid line.
3. Which side to shade? Since it says \(y > \dots \), we shade the region above the line. If you aren't sure, pick a "test point" like \((0,0)\) and see if it makes the inequality true!
Quadratic Graphs (e.g., \(y \le ax^2 + bx + c\))
This works exactly the same way. Draw the parabola (solid or dashed) and then shade above for \( > \) or below for \( < \).
Common Mistake to Avoid:
Students often forget to check if the line should be dashed or solid. Always look at the sign first! No "equal to" bar means the line is dashed.
Key Takeaway: A graph is just a picture of all the possible correct answers. Use solid lines for "equal to" and shade the correct side by looking at the \(y\) value.
Summary Checklist
• Can I solve linear inequalities and remember to flip the sign for negatives?
• Can I find critical values for a quadratic and sketch the curve?
• Do I know when to use \(\cup\) (OR) and \(\cap\) (AND)?
• Can I switch between \( x > 3 \) and interval notation \( (3, \infty) \)?
• Can I represent \(y > f(x)\) on a graph with the correct shading and line type?
If you can do these, you've mastered AS Level Inequalities! Great job!