🧮 algebra

1. **State the problem:** We are given input-output pairs: inputs $5,6,7,8$ and outputs $11,14,17,20$. We want to find the expression (function) that relates the input $x$ to the o

1. The problem is to solve the compound inequality $-2x > -5 > -15$. 2. Compound inequalities can be split into two separate inequalities: $-2x > -5$ and $-5 > -15$.

1. **Problem:** Find an equation whose roots are $a^3\beta$ and $\alpha\beta^3$, where $a$ and $\beta$ are roots of $2x^2 - 4x + 1 = 0$. 2. **Step 1: Identify sum and product of ro

1. **State the problem:** Simplify the expression $$(5x-3)(x+1) - 6x + 10x^2 + (3-5x)^2$$. 2. **Recall formulas and rules:**

1. **State the problem:** Simplify the expression $$(5x-3)(x+1)-6x+10x^2+(3-5x)^2$$. 2. **Recall formulas and rules:**

1. **Problem statement:** We need to find how many students are selected in total for the competition if 20% of the boys and 25% of the girls from all classes are selected. 2. **St

1. مسئله را بیان می‌کنیم: مجموع دو عدد برابر با $\frac{45}{12}$ است و یکی از اعداد $\frac{11}{3}$ واحد بزرگتر از دیگری است. باید عدد بزرگتر را پیدا کنیم. 2. فرض کنیم عدد کوچکتر $x$

1. **State the problem:** Solve the equation $6 + 2x = x - 6$ for $x$. 2. **Write down the equation:**

1. Let's start by understanding the problem: you want a detailed explanation of equations, why we use them, and the reasoning behind each step. 2. Equations are mathematical statem

1. **Problem statement:** Simplify the expression $\frac{3}{2} + \frac{5}{2} \div \frac{5}{4}$. 2. **Recall the division of fractions rule:** Dividing by a fraction is the same as

1. **State the problem:** Solve the equation $3y + 15 = 5y + 1$ for $y$. 2. **Write down the equation:**

1. **State the problem:** Solve the equation $$30000x + 67500y + 30000xy = 39375$$ for one variable in terms of the other. 2. **Rewrite the equation:** Group terms involving $x$:

1. **State the problem:** Solve the equation $2m + 5 = 10$ for $m$. 2. **Formula and rules:** To solve for $m$, isolate the variable by performing inverse operations. Subtract 5 fr

1. The user asks for an answer in a style that helps them learn how to do it themselves. 2. Since no specific math problem is given, I will provide a general example of solving a s

1. **Problem Statement:** Solve the system of linear equations: $$\begin{cases} 3x_1 + x_2 + x_3 = 8 \\ x_1 + 4x_2 + 2x_3 = 15 \\ 2x_1 + x_2 + 5x_3 = 19 \end{cases}$$

1. **State the problem:** Solve the equation $8k - 1 = 15$ for $k$. 2. **Write the formula and rules:** To solve for $k$, isolate $k$ on one side of the equation by performing inve

1. **State the problem:** Find the inverse function of $f(x) = 7x^3 - \frac{2}{20}$ where $x \in \mathbb{R}$. 2. **Rewrite the function:** Simplify the constant term: $f(x) = 7x^3

1. The problem is to analyze the function $f(x) = \frac{10}{7x^3 - 2}$ where $x \in \mathbb{R}$. 2. This is a rational function where the numerator is a constant 10 and the denomin

1. The problem asks to find the inverse function of \( f(x) = \frac{10}{7x^3} - 2 \) where \( x \in \mathbb{R} \). 2. To find the inverse function \( f^{-1}(x) \), we start by sett

1. **State the problem:** We need to find the mean daily wage of all workers in the company. 2. **Given data:**

1. **State the problem:** Find the equation of the tangent line to the circle given by $$x^2 + y^2 + 14x + 18y - 39 = 0$$ at the point in the second quadrant where $$x = -2$$. 2. *