🧮 algebra

1. **State the problem:** We need to determine which expressions always yield an even value when $x$ is an odd number. 2. **Recall important rules:**

1. **State the problem:** We are given the equation $$f = \frac{b^2 + w}{2 - b}$$ and need to find the value of $w$ when $b = 4$ and $f = -10$. 2. **Write down the formula:** $$f =

1. **State the problem:** We have a system of three equations: $$kx + 3y - z = 3$$

1. **State the problem:** We need to find the value of $w^r$ when $w=5$ and $r=3$. 2. **Formula:** The expression $w^r$ means $w$ raised to the power of $r$, which is multiplying $

1. **State the problem:** We need to find the value of $6 + y^2$ when $y = 3$. 2. **Formula used:** The expression is $6 + y^2$.

1. **State the problem:** Solve for $x$ in the equation $4x = 13x - 18$. 2. **Rewrite the equation:** Move all terms involving $x$ to one side to isolate $x$.

1. **State the problem:** Solve the equation $4x - 13x = -38$ for $x$. 2. **Combine like terms:** On the left side, combine $4x$ and $-13x$.

1. The problem is to solve the inequality $\ln x \geq -1$. 2. Recall that the natural logarithm function $\ln x$ is defined only for $x > 0$.

1. **State the problem:** Simplify or factor the polynomial $2x^3 - 3x^2 - 3x + 2$. 2. **Recall the factoring approach:** For cubic polynomials, try to find rational roots using th

1. **State the problem:** We are comparing the slopes of two linear functions, Function A and Function B. 2. **Given information:**

1. **State the problem:** We have two linear functions, Function A: $y=4x+3$ and Function B, which passes through points near $(-10,-10)$ and $(10,10)$, indicating it is the line $

1. The problem is to find the slope $m$ of the line passing through the points $(-7,7)$ and $(7,-7)$. 2. The formula for slope between two points $(x_1,y_1)$ and $(x_2,y_2)$ is:

1. **Problem:** Write an equation that represents the total cost $y$ for $x$ GB of data given a fixed fee of 10 plus 3 per GB. 2. **Formula:** The total cost $y$ can be expressed a

1. **State the problem:** Simplify the expression $$\left[\sqrt{s-1} + \ln \sqrt{s^{2} - 1^{2}}\right]^{-1}$$. 2. **Rewrite the expression inside the logarithm:** Note that $$1^{2}

1. Stating the problem: Solve the equation $\log_{2x}(4) = \log_x(2)$.\n\n2. Recall the change of base formula for logarithms: $\log_a b = \frac{\log b}{\log a}$. We will use natur

1. **State the problem:** Solve the quadratic equation $$10x^2 + x + 3 = 6$$ using the quadratic formula. 2. **Rewrite the equation in standard form:** Move all terms to one side:

1. **State the problem:** Solve the quadratic equation $$4q^2 + q + 1 = -3q$$ using the quadratic formula. 2. **Rewrite the equation in standard form:** Move all terms to one side:

1. **Постановка задачі:** Дано натуральне число $n > 1$. Потрібно знайти найменшу кількість натуральних чисел, факторіали яких у сумі дорівнюють $n! - 1$.

1. The problem is to understand the ratio $7 : 5 : 8$ and what it represents. 2. A ratio compares quantities to each other, showing their relative sizes.

1. **State the problem:** Solve the equation $$\sqrt{x} - 2 = \sqrt{x} 2x + 1$$. 2. **Rewrite the equation for clarity:** The equation is $$\sqrt{x} - 2 = 2x\sqrt{x} + 1$$.

1. **State the problem:** Simplify the expression $\frac{41}{49} \div \frac{1}{2}$. 2. **Recall the division rule for fractions:** Dividing by a fraction is the same as multiplying