2. Algebra

2.1 Linear Inequations (in one variable)

✦ Explanation: ● An inequation is a mathematical sentence expressing the inequality between two expressions. ● The solution of an inequation is the set of values that satisfy the inequality. ● Solving involves treating it like an equation, except inequality signs must be preserved (or reversed when multiplying/dividing by negative numbers). ✦ Example: Solve: 3x−5<16 Solution: Add 5: 3x < 21 Divide by 3: x < 7 Solution set: All real numbers less than 7.

Linear inequations are inequalities that can contain one variable. To solve them, you manipulate the equation similar to how you would with an equality, but you must remember the following important rules: If you multiply or divide both sides of the inequation by a negative number, you have to reverse the inequality sign. For example, in the inequation 3x - 5 < 16, we first add 5 to both sides, giving us 3x < 21. Then, we divide by 3, yielding x < 7. Thus, the solution set is all real numbers less than 7.

Imagine you have a budget of $100 for a party, and each item you want to buy costs$x. The total spending must be less than $100. You could express this as the inequality: x < 100. This means you can only spend less than that amount. If you decide on the price of certain items, you can find the maximum amount you could spend on one item!

2.2 Quadratic Equations

✦ Explanation: ● A quadratic equation is in the form: ax²+bx+c=0,(a≠0) ● It can be solved using: ○ Factorization ○ Quadratic formula: x=−b±√(b²−4ac)/(2a) ✦ Example: Solve: x²−5x+6=0 Solution: Factor: (x−2)(x−3)=0 ⇒ x = 2 or x = 3

A quadratic equation is a polynomial equation of degree 2, usually written in the form ax² + bx + c = 0, where a, b, and c are constants, and a cannot be zero. Quadratic equations can be solved by factoring, using the quadratic formula, or completing the square. In the example x² - 5x + 6 = 0, we factor it into (x - 2)(x - 3) = 0, which gives us the solutions x = 2 and x = 3.

Think of a garden shaped like a parabolic arch where the width and height is defined by the equation of the garden's boundary. If you want to know the points at which the garden meets a specific height (like the ground level), you're actually solving a quadratic equation that describes the height of the arch at various widths.

2.3 Factorisation of Polynomials

✦ Explanation: ● Factorising a polynomial means writing it as a product of its factors. ● Common methods: ○ Taking out common factors ○ Splitting the middle term ○ Using identities like: (a+b)²=a²+2ab+b² ✦ Example: Factorise: x²+7x+10 Solution: Split middle term: x²+5x+2x+10=x(x+5)+2(x+5)=(x+2)(x+5)

Factorisation is the process of expressing a polynomial as a product of simpler polynomials (factors). One common method is to look for common factors among the terms. Another method can involve splitting the middle term or applying known identities. For instance, in x² + 7x + 10, we can split it into terms that multiply to 10 (the constant term) and add to 7 (the coefficient of x). This results in (x + 2)(x + 5).

Imagine you have a box of chocolates, and you want to divide them into smaller packages. You can see how many chocolates you have and find different ways to distribute them based on the total. Each way you divide them represents a factorization of the total amount into smaller groups.

2.4 Ratio and Proportion

✦ Explanation: ● A ratio compares two quantities: a:b=ab ● Proportion means equality of two ratios: If a:b=c:d, then a, b, c, d are in proportion. ✦ Example: If 2:3 = x:6, find x. Solution: 2/3=x/6⇒x=2×6/3=4

Ratios are a way to compare two quantities, and they can be written in the form a:b. When two ratios are equal, they are said to be in proportion. For example, in the proportion 2:3 = x:6, we can cross-multiply to find the value of x, leading us to x = 4.

Picture a recipe that calls for 2 cups of flour for every 3 cups of sugar. If you want to keep that same ratio, but you're cooking for more people, you would scale up the amounts. Understanding ratios helps ensure you keep the taste consistent, no matter how large the batch!

2.5 Matrices (Order 2 × 2)

✦ Explanation: ● A matrix is a rectangular array of numbers. ● A 2 × 2 matrix has 2 rows and 2 columns. ● You can add, subtract, and multiply matrices (only when orders match appropriately). ✦ Example: Let A=[1,2;3,4], B=[5,6;7,8] Find A + B. Solution: A+B=[1+5, 2+6; 3+7, 4+8] = [6, 8; 10, 12]

A matrix is a collection of numbers arranged in rows and columns. A 2 × 2 matrix has two rows and two columns, allowing for mathematical operations like addition, subtraction, and multiplication. When adding two matrices, add the corresponding entries. For example, to add matrices A and B given above, we would perform (1+5), (2+6), and so forth, resulting in a new matrix.

Think of a matrix like a seating chart for a small theater where each seat can hold certain individuals. Each row and column could represent a specific type of seating in the theater. If you want to know how many people attended an event, you can quickly visualize it with a matrix rather than counting individuals one by one.

2.6 Arithmetic Progression (A.P.)

✦ Explanation: ● A sequence where the difference between consecutive terms is constant. ● nth term: an=a+(n−1)d ● Sum of n terms: Sn=n/2[2a+(n−1)d] ✦ Example: Find the 10th term of the A.P.: 5, 8, 11, … Solution: a = 5, d = 3 an=5+(10−1)×3=5+27=32

An arithmetic progression (A.P.) is a sequence of numbers where each term is formed by adding a constant difference (d) to the previous term. The nth term formula allows us to find any term in the sequence, and the sum formula helps us compute the total of the first n terms. For instance, in the sequence 5, 8, 11, the first term (a) is 5, and since each term increases by 3 (the common difference), we can calculate the 10th term.

Consider a bus schedule where a bus leaves the station every 10 minutes. If the first bus departs at 8:00 AM, the times at which the buses leave form an arithmetic progression: 8:00 AM, 8:10 AM, 8:20 AM, and so on. This pattern helps passengers know exactly when to expect the next bus!

2.7 Geometric Progression (G.P.)

✦ Explanation: ● A sequence where each term is obtained by multiplying the previous term by a constant (common ratio). ● nth term: an=a⋅r^(n−1) ● Sum of first n terms (when r ≠ 1): Sn=a⋅(r^n−1)/(r−1) ✦ Example: Find the 5th term of a G.P. with a = 2, r = 3 Solution: an=2×3^4=2×81=162

A geometric progression (G.P.) is a sequence where each term is generated by multiplying the previous term by a fixed ratio (r). The nth term can be calculated using the formula a * r^(n−1). For example, if your first term (a) is 2 and the common ratio (r) is 3, the 5th term can be computed easily using the formula, resulting in 162.

Imagine saving money in a bank where your savings double every year. If you start with $2, in the first year you have$2, in the second year $4, in the third year$8, and so forth. This quick growth showcases how a geometric progression works in financial contexts!

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