Inequalities: Meaning and Definitions

Inequalities in mathematics are statements that compare two values, showing that one is greater than, less than, or not equal to the other. Unlike equations which assert equality, inequalities describe a range of possible values. Think of it like a see-saw: equations show a balanced see-saw, while inequalities show one side higher or lower than the other.

➕ Basic Inequality Symbols

• >: Greater than (e.g., 5 > 3 means 5 is greater than 3)
• <: Less than (e.g., 2 < 7 means 2 is less than 7)
• ≥: Greater than or equal to (e.g., x ≥ 4 means x can be 4 or any number larger than 4)
• ≤: Less than or equal to (e.g., y ≤ 10 means y can be 10 or any number smaller than 10)
• ≠: Not equal to (e.g., a ≠ b means a is not the same value as b)

💡Real-World Analogy

Imagine you're setting a budget for your weekly groceries. You might say your spending, represented by 's', must be ≤ $50. This means you can spend exactly $50 or any amount less than that, but not more. This is a common real-world example of using inequalities!

🔑 Properties of Inequalities

• Addition/Subtraction Property: Adding or subtracting the same number from both sides of an inequality does not change the inequality. For example, if x > y, then x + 2 > y + 2.
• Multiplication/Division Property: Multiplying or dividing both sides by the same positive number does not change the inequality. If x < y and z > 0, then xz < yz.
• Important Caveat: Multiplying or dividing both sides by a negative number reverses the inequality. If x < y and z < 0, then xz > yz. This is a crucial rule to remember!

✍️ Solving Inequalities

Solving inequalities is similar to solving equations, but with the added consideration of the negative number rule. The goal is to isolate the variable on one side of the inequality.

1. Simplify both sides of the inequality.
2. Use addition/subtraction to move terms around.
3. Use multiplication/division to isolate the variable, remembering to reverse the inequality sign if multiplying or dividing by a negative number.

Key Takeaway: Inequalities provide a powerful tool for describing ranges and constraints, unlike equations which focus on exact values. Always remember to flip the sign when multiplying or dividing by a negative number!