🧮 algebra

1. **Stating the problem:** We need to evaluate the summation $$\sum_{i=5}^{6} i$$ which means adding all integers from $i=5$ to $i=6$ inclusive. 2. **Formula and explanation:** Th

1. **State the problem:** We need to evaluate the summation $$\sum_{i=1}^{6} i$$ and compare it to the value 5. 2. **Recall the formula for the sum of the first n natural numbers:*

1. **State the problem:** Evaluate the logarithmic expression $$\log_2(4)$$. 2. **Recall the definition of logarithm:** $$\log_b(a) = c$$ means $$b^c = a$$.

1. **State the problem:** Find all vertical and horizontal asymptotes of the function $$f(x) = \frac{-3x^2 - 14x + 49}{4x^2 - 9x + 2}.$$ 2. **Vertical asymptotes:** These occur whe

1. **State the problem:** We need to evaluate the function $$f(x) = \frac{x^2}{x - 4}$$ at $$x = 4$$. 2. **Substitute the value:** Replace $$x$$ with $$4$$ in the function:

1. **State the problem:** Solve the equation $$\frac{11x-33}{x-3} = x$$ and verify the solution $x=11$. 2. **Recall the formula and rules:** The equation involves a rational expres

1. **State the problem:** Solve the equation $$\frac{11x-33}{x-3} = x$$ for $x$. 2. **Recall the formula and rules:** To solve rational equations, multiply both sides by the denomi

1. **Problem statement:** Given the function $$f(x) = \frac{x - 1}{x^2 - 1} - 2x,$$ find:

1. **State the problem:** Calculate $850 + 150\%$ of $850$. 2. **Formula and explanation:** To add a percentage of a number to the number itself, use the formula:

1. **State the problem:** We need to find the constant of proportionality $k$ from the graph showing the cost of apples versus the number of apples. 2. **Understand the relationshi

1. **State the problem:** We are given a graph showing the cost of apples versus the number of apples bought. We need to identify a point on the graph, write a "per" statement, and

1. **Problemstellung:** Marie hat ein Handy, dessen Akku zu Beginn zu $18\%$ gefüllt war und nach $1 \frac{1}{2}$ Stunden (also $1.5$ Stunden) zu $84\%$. Der Aufladevorgang ist lin

1. 题目要求判断下列二次根式中哪个是最简二次根式。 2. 最简二次根式的定义是：根号内没有可以开平方的因数（除了1），即根号内的数不能再被分解成平方数的乘积。

1. The problem asks why we use $5x$ and $3x$ in algebraic expressions. 2. In algebra, $x$ represents a variable, and the numbers in front of $x$ are coefficients.

1. **State the problem:** We are given two numbers whose total sum is $112$ and their ratio is $5:3$. We need to find the two numbers. 2. **Formula and rules:** If two numbers are

1. The problem is to simplify the expression $20X3$. 2. This expression represents multiplication of three numbers: 20, X, and 3.

1. **Planteamiento del problema:** Queremos encontrar escalares $$\alpha_1$$, $$\alpha_2$$ y $$\alpha_3$$ tales que:

1. **Planteamiento del problema:** Queremos saber si la matriz $$M = \begin{pmatrix}-7 & -5 \\ -3 & 4\end{pmatrix}$$ puede expresarse como combinación lineal de las matrices $$A =

1. The problem is to evaluate the expression $10^^^^7$. 2. The notation $^^^^$ is not standard in mathematics. However, it is often used to denote tetration, which is iterated expo

1. **Problem:** Identify the inequality graphed on a number line with an open circle at $-3$ and a shaded ray extending to the right. 2. **Understanding the graph:**

1. **Problem:** Find how many units to the right is from $-5$ to $+5$ on a number line. 2. **Formula:** The distance between two points $a$ and $b$ on a number line is given by the