🧮 algebra

1. Problem statement: Find the constant of variation $k$ if $y$ varies inversely as $x$, and $y = 12$ when $x = 9$. Since $y$ varies inversely as $x$, the relation is:

1. **Simplify the surds:** (a)(i) Simplify $\sqrt{2} + 3\sqrt{3} - 5\sqrt{2} + \sqrt{3}$.

1. **Problem statement:** Prove by mathematical induction that $$\sum_{i=1}^{n} \frac{1}{i(i+1)} = \frac{n}{n+1}$$

1. **State the problem:** The volume $V$ of a gas is inversely proportional to its pressure $P$. If the volume is $360$ cm³ when the pressure is $20$ g/cm³, find the pressure when

1. **Problem statement:** Illustrate a real-life example of an inverse variation and solve multiple problems applying the concept. **Example:** The time needed to complete a job va

1. The problem states that $f(x) = ax^2 + 2x - 11$ and the point $(1, 0)$ lies on the graph of $f$. This means when $x = 1$, $f(x) = 0$. 2. Substitute $x = 1$ and $f(1) = 0$ into t

1. The problem involves a pattern of hexagonal numbers with each top-right number related to the numbers in the left columns and the middle numbers. 2. To find the rule, observe ea

1. The problem is to solve the equation $$|2x + 5| = 9$$ for $$x$$. 2. Absolute value equations mean the expression inside can be either positive or negative but the result is posi

1. The problem asks us to find the domain of the rational function $$f(x) = \frac{5}{x - 6}$$ and understand its graph. 2. The domain of a rational function includes all real numbe

1. **State the problem:** Simplify the expression $$3^{n+2} + \left(3^{n+3} - 3^{n+1}\right)$$ and determine which choice among (A), (B), (C), or (D) it equals. 2. **Rewrite terms

1. **State the problem:** We need to solve the inequality $4x + 2 > 16$ and determine which of the multiple-choice answers is equivalent. 2. **Solve the inequality:**

1. **State the problem:** Find the domain of the function $$h(x) = \sqrt{\frac{1 - x}{x + 1}}$$ which means we need all values of $$x$$ for which the expression under the square ro

1. **State the problem:** We analyze the function $f(x) = \frac{5}{x - 6}$. 2. **Identify the domain:** The function is a rational expression and is undefined when the denominator

1. The problem asks us to identify the equation of a polynomial $p(x)$ based on the shape and roots of its graph. 2. The polynomial's roots are at $x = 0$, $x = 3$, and $x = -\frac

1. Stating the problem: Simplify or factor the quadratic expression $x^2 + 4x + 3$. 2. To factor, we look for two numbers whose product is $3$ (the constant term) and whose sum is

1. **Problem 1: Find the general equation of the hyperbola with vertices (0, ±2) and foci (0, ±2\sqrt{5})**. Step 1: Identify the orientation and parameters.

1. **State the problem:** We want to simplify the expression $$\log \left( \sqrt{\frac{7^2 t^3 p}{d^6 b^2}} \right)$$. 2. **Rewrite the square root as an exponent:** Recall that $$

1. We are given the equation $x + \frac{1}{x} = 3$ and need to find the value of $x^{15} + \frac{1}{x^{15}}$. 2. Let's denote $a_n = x^n + \frac{1}{x^n}$. We know $a_1 = 3$.

1. Problem: Simplify the expression $$\log\left(\sqrt{\frac{7^2 t^3 p}{d^t b^2}}\right)$$. 2. First, rewrite the square root as a fractional exponent: $$\sqrt{x} = x^{1/2}$$, so

1. Simplify the expressions. 1.a. Simplify $3(x - y) - 3(2x + 3y)$

1. **State the problem:** We have a rectangular garden that is 600 m long and 300 m wide, and we need to calculate the cost of fencing around it. 2. **Calculate the perimeter of th