Hello, Grade 9 students! Let's get to know "Inequalities."
In everyday life, we often encounter situations that aren't always about being "equal." For example, "You need at least 50 baht to buy this snack," or "This elevator can carry no more than 500 kilograms." These statements are the origin of Inequalities in mathematics.
In this chapter, we will learn how to solve problems when two things are "not equal." If you feel like math is difficult, don't worry! We will walk through it together step-by-step with tips to help you remember the concepts easily.
1. Inequality Symbols You Need to Know
First, we need to know the "signs" used to describe these relationships:
- \( < \) means less than
- \( > \) means greater than
- \( \leq \) means less than or equal to (no more than)
- \( \geq \) means greater than or equal to (at least)
- \( \neq \) means not equal to
Memory Tip: Imagine the \( < \) or \( > \) sign as a "crocodile mouth." The hungry crocodile always opens its mouth wide to chomp on the side with the "greater" value!
Important points that are often confusing:
The phrase "no more than" means it can be that value or less, so use \( \leq \).
The phrase "at least" means it can be that value or more, so use \( \geq \).
2. Reading and Graphing Solutions
The solution to an inequality often includes many values (a range), so we use a number line to show the answer. There are two points to observe:
- Open circle (empty white dot): Used with \( <, > \), and \( \neq \) (meaning exclude that number).
- Closed circle (solid black dot): Used with \( \leq \) and \( \geq \) (meaning include that number).
Example:
\( x > 3 \) means all numbers greater than 3 (draw a line to the right from an open circle at 3).
\( x \leq 5 \) means all numbers less than or equal to 5 (draw a line to the left from a closed circle at 5).
Did you know? A linear inequality in one variable is an inequality that has only one variable (like \( x \)), and the exponent of that variable is exactly 1.
3. How to Solve Inequalities (Step-by-Step)
Solving inequalities is very similar to solving equations (finding the value of \( x \)). We need to isolate the variable \( x \) on one side.
Addition and Subtraction Rules:
You can add or subtract the same value from both sides and the inequality sign remains the same.
Multiplication and Division Rules (The most important part!):
1. If multiplying or dividing by a positive number: The sign stays the same.
2. If multiplying or dividing by a negative number: You must "reverse the sign" to the opposite direction!
Example 1: Solve the inequality \( x + 5 < 12 \)
1. We want \( x \) by itself.
2. Move \( +5 \) to the other side by subtracting it from both sides.
3. We get \( x < 12 - 5 \)
4. The solution is \( x < 7 \)
Example 2: (Case where you must reverse the sign) Solve the inequality \( -2x \leq 10 \)
1. We need to divide both sides by \( -2 \).
2. Be careful! Because we are dividing by a negative number (\(-2\)), we must change \( \leq \) to \( \geq \).
3. We get \( x \geq \frac{10}{-2} \)
4. The solution is \( x \geq -5 \)
Common Mistake: Many students forget to flip the sign when multiplying or dividing by a negative number. Remember: "Whenever you multiply or divide by a negative, flip the sign immediately!"
4. Solving Inequality Word Problems
When you see a long word problem, stay calm and follow these steps:
- Define the variable: Let \( x \) represent what the question is asking for.
- Translate into a symbolic expression: Look for keywords like "less than," "at most," "not less than."
- Solve the inequality: Use the isolation methods we learned.
- Check the answer: See if the result makes sense in the context of the problem.
Example Problem: Mana has some money. His mom gives him another 100 baht, making his total money not less than 250 baht. At least how much money did Mana have initially?
Reasoning:
- Let \( x \) represent Mana's initial money.
- Write the inequality: \( x + 100 \geq 250 \)
- Isolate the variable: \( x \geq 250 - 100 \)
- Solution: \( x \geq 150 \) (Mana had at least 150 baht originally).
Key Takeaways
- Signs: Remember the crocodile mouth; it always opens toward the larger value.
- Graphs: An open circle means the number is not included; a closed circle means the number is included.
- The Golden Rule: Multiplying or dividing by a "negative number" means you must "reverse the sign."
- Word Problems: Patiently translate words into mathematical symbols.
Closing thought: Inequalities might seem a bit fiddly with the signs at first, but if you keep practicing, you'll naturally start to see when to flip them. Keep going—I believe you can definitely do it!