🧮 algebra

1. **Problem statement:** We have a function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = x + 1$. We need to check if $f$ is injective, surjective, or bijective. 2. **Definiti

1. **Problem statement:** We have a function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = x + 1$. We need to check if $f$ is injective, surjective, or bijective. 2. **Definiti

1. **State the problem:** A farmer stocked a pond with 1000 fish fingerlings. The fish population grows by a constant factor $x$ each year. After 2 years, the population is 36000.

1. **Problem statement:** We have a function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = x + 1$. We need to check if $f$ is injective, surjective, or bijective. 2. **Definiti

1. **State the problem:** Given $x = 3 + \sqrt{8}$, find the value of $x - \frac{1}{x}$. 2. **Recall the formula and rules:** To find $x - \frac{1}{x}$, we need to calculate $\frac

1. **State the problem:** Given $x = 3 + \sqrt{8}$, find the value of $x - \frac{1}{x}$. 2. **Recall the formula:** To find $x - \frac{1}{x}$, we can use the expression directly by

1. Let's solve a challenging factorial problem: Calculate $$\frac{10!}{8!}$$. 2. Recall the factorial definition: $$n! = n \times (n-1) \times (n-2) \times \cdots \times 1$$.

1. The problem asks why $10! = 10 \times 9 \times 8!$. 2. The factorial of a number $n$, written as $n!$, is the product of all positive integers from $n$ down to 1.

1. The definite integral from 3 to 10 of $x$ dx is calculated using the formula for the integral of $x$, which is $\frac{x^2}{2}$. Evaluating from 3 to 10: $$\int_3^{10} x \, dx =

1. The problem is to create a challenging factorial problem for practice. 2. Factorials are denoted by $n!$ and represent the product of all positive integers from 1 to $n$.

1. The problem is to solve a system of equations with 1, 2, and 3 unknowns. 2. For 1 unknown, the equation is usually of the form $ax = b$. To solve, divide both sides by $a$ to ge

1. **Problem Statement:** Find Shabnam's age when Aftab's age is 23 years, given Shabnam is 3 years older than Aftab. 2. **Formula:** Shabnam's age $s = a + 3$, where $a$ is Aftab'

1. **Problem:** A forester wants to plant 68 apple trees and 110 mango trees in equal rows (same number of trees per row). Each row contains only one type of tree. Find the minimum

1. **Problem:** Find the factor of $(a + 5)(a - 9) - 15$ from the given options. 2. **Formula and rules:** Use distributive property to expand and simplify expressions.

1. **Problem:** Find the factor of $(a + 5)(a - 9) - 15$ from the given options. 2. **Formula and rules:** Expand the product and simplify the expression.

1. **State the problem:** Convert a quadratic function from general form $y = ax^2 + bx + c$ to standard (vertex) form $y = a(x-h)^2 + k$. 2. **Formula and rules:** The vertex form

1. The problem is to solve the equation: $0 = 0$. 2. This equation states that zero equals zero, which is always true.

1. **Menyatakan masalah:** Kita diberikan persamaan Diophantine $x + y + z = n$ dengan $n \in \mathbb{N}$ dan diminta menentukan solusi untuk $n=10$ dalam tiga kasus: (a) solusi bi

1. **Menyatakan masalah:** Kita diberikan persamaan Diophantine $x + y + z = n$ dengan $n \in \mathbb{N}$ dan diminta mencari solusi untuk $n=10$. 2. **Aturan dasar:**

1. **State the problem:** Simplify the expression $\sqrt{3} + \sqrt{5}$. 2. **Understand the terms:** $\sqrt{3}$ and $\sqrt{5}$ are square roots of 3 and 5 respectively. They are i

1. The problem is to simplify the fraction $\frac{600}{800}$. 2. The formula to simplify a fraction is to divide the numerator and denominator by their greatest common divisor (GCD