Hey guys! Welcome to your ultimate guide for Grade 10 Pure Maths Term 1. We're going to break down everything you need to know, making it super easy to understand and ace those exams. Let's dive in!

1. Algebra Essentials

Algebra forms the bedrock of much of what you'll learn in Pure Maths. Mastering it early on is super important. So, let's go through the key concepts you need to nail.

1.1 Algebraic Expressions

First off, let's talk about algebraic expressions. These are combinations of variables, constants, and operations. Think of them as the sentences of the math world. For example, 3x + 5, 2y^2 - x + 7, and ab/c are all algebraic expressions. Simplifying these expressions is a fundamental skill. It involves combining like terms and applying the order of operations (PEMDAS/BODMAS).

Why is this important? Understanding how to manipulate algebraic expressions allows you to solve more complex problems later on. It's like learning the alphabet before writing a novel – you can't skip it!

Combining Like Terms: Like terms are terms that have the same variable raised to the same power. For example, 3x and 5x are like terms, but 3x and 5x^2 are not. When combining like terms, you simply add or subtract their coefficients. So, 3x + 5x becomes 8x. Simple, right? Now, let's throw in some numbers: 7y - 2y + 4 + 6. Combine the y terms to get 5y and the constants to get 10. So, the simplified expression is 5y + 10.

Order of Operations (PEMDAS/BODMAS): This is your golden rule! It tells you the order in which to perform operations: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). For example, in the expression 2 + 3 * 4, you first multiply 3 * 4 to get 12, then add 2 to get 14. If you did it the other way around, you'd get 20, which is wrong! Remember, PEMDAS/BODMAS isn't just a suggestion; it's the law!

Practice Makes Perfect: The more you practice simplifying algebraic expressions, the better you'll get. Start with simple expressions and gradually work your way up to more complex ones. Do plenty of exercises, and don't be afraid to make mistakes. Mistakes are learning opportunities in disguise!

1.2 Equations and Inequalities

Next up, we tackle equations and inequalities. An equation states that two expressions are equal, while an inequality shows that one expression is greater than, less than, greater than or equal to, or less than or equal to another.

Equations: Solving equations involves finding the value(s) of the variable(s) that make the equation true. The most common type is a linear equation, like 2x + 3 = 7. To solve this, you want to isolate x on one side of the equation. Subtract 3 from both sides to get 2x = 4, then divide by 2 to get x = 2. Voila! You've solved for x.

Inequalities: Solving inequalities is similar to solving equations, but with one key difference: when you multiply or divide both sides by a negative number, you must flip the inequality sign. For example, consider the inequality -3x < 9. Divide both sides by -3, and remember to flip the sign! You get x > -3. This means that any value of x greater than -3 will satisfy the inequality.

Graphing Inequalities: Inequalities can also be represented graphically on a number line. For example, to graph x > -3, you would draw an open circle at -3 (because -3 is not included) and shade everything to the right. If the inequality was x ≥ -3, you would use a closed circle at -3 to indicate that -3 is included.

1.3 Factorization

Factorization is the process of breaking down an algebraic expression into its constituent factors. It's like reverse engineering. For example, the expression x^2 + 5x + 6 can be factored into (x + 2)(x + 3). This is super useful for solving quadratic equations and simplifying expressions.

Common Techniques: There are several common factorization techniques. The first is finding the greatest common factor (GCF). For example, in the expression 4x^2 + 8x, the GCF is 4x. Factoring this out gives you 4x(x + 2). Another common technique is using the difference of squares formula: a^2 - b^2 = (a + b)(a - b). For example, x^2 - 9 can be factored into (x + 3)(x - 3).

Quadratic Trinomials: Factoring quadratic trinomials like x^2 + 5x + 6 involves finding two numbers that multiply to the constant term (6 in this case) and add up to the coefficient of the x term (5 in this case). These numbers are 2 and 3, so the factored form is (x + 2)(x + 3). Practice spotting these patterns, and you'll become a factorization pro in no time!

2. Functions and Graphs

Functions and graphs are all about relationships – how one thing changes in relation to another. Let's get you familiar with the basics.

2.1 Introduction to Functions

A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. Think of it as a machine: you put something in (the input), and it spits something out (the output). For example, the function f(x) = 2x + 1 takes an input x, multiplies it by 2, and adds 1 to get the output. If you input x = 3, the output is f(3) = 2(3) + 1 = 7.

Domain and Range: The domain of a function is the set of all possible input values, and the range is the set of all possible output values. For example, if f(x) = √x, the domain is all non-negative real numbers (because you can't take the square root of a negative number), and the range is also all non-negative real numbers.

Function Notation: Function notation can seem a bit confusing at first, but it's actually quite simple. f(x) just means