🧮 algebra

1. **State the problem:** We have a linear function $f(x)$ passing through points $(2,5)$ and $(-1,10)$. We need to find the slope of a line perpendicular to $f(x)$. 2. **Formula f

1. **State the problem:** We need to find all values of $x$ such that $$f(x) = \log_2 x + \log_2 (x - 2) = 3.$$ 2. **Recall the logarithm property:** The sum of logarithms with the

1. **Stating the problem:** We are given two functions related to sound intensity and decibels: $$d(Y) = 10 \log_{10}(Y \cdot 10^{12})$$

1. **State the problem:** Given the arithmetic progression (A P) 3, x, y, 18, find the values of $x$ and $y$. 2. **Recall the formula for an arithmetic progression:** The differenc

1. **State the problem:** Simplify the expression $$\frac{2}{3}(21x + 15) + 7 + 8 + 3x + 1$$. 2. **Use the distributive property:** Multiply $$\frac{2}{3}$$ by each term inside the

1. **State the problem:** Simplify the expression $\left(5 \frac{7}{3}\right) \times \left(5 \frac{8}{3}\right)$ and express it in the form $5k$ where $k$ is a number. 2. **Convert

1. **State the problem:** A magician can now hold his breath for 41 seconds, which is 0% more than when he first tried. We need to find how long he could hold his breath originally

1. **State the problem:** The PTA raised 16750 this year, which is 0% less than last year. We need to find how much was raised last year.

1. Сформулюємо задачу: знайти точки екстремуму функції $$y = x^4 - 2x^2 + 1$$. 2. Для знаходження точок екстремуму потрібно знайти похідну функції та прирівняти її до нуля, оскільк

1. **State the problem:** Solve the equation $$3| -10 + x | = 21$$ for $x$. 2. **Recall the absolute value equation rule:** For $|A| = B$, where $B \geq 0$, the solutions are $A =

1. **Problem:** Find the absolute value of the complex number $2 - 10i$. 2. **Formula:** The absolute value (or modulus) of a complex number $a + bi$ is given by

1. **State the problem:** Find the equation of the line passing through points $(3, -4)$ and $(-2, 6)$ in gradient-intercept form $y = mx + b$. 2. **Formula for slope:** The slope

1. **Problem:** Solve the inequality $$x^4 + 35x^2 - 36 \geq 0$$. 2. **Formula and rules:** To solve polynomial inequalities, first find the roots by setting the expression equal t

1. **State the problem:** Solve the equation $$-12 = -2(2x - 5) - 22$$ for $x$. 2. **Use the distributive property:** Multiply $-2$ by each term inside the parentheses:

1. **State the problem:** Calculate the value of $$\sqrt[3]{81} + 8^{\frac{1}{3}}$$. 2. **Recall the cube root and exponent rules:** The cube root of a number $a$ is the same as $a

1. The problem asks to write an inequality for the situation where the number of days, $d$, of rain is not 17. 2. The key phrase here is "not 17," which means $d$ can be any number

1. The problem states: A minimum of 36 people, $p$, must ride in the bus. 2. To write an inequality for this situation, we need to express that the number of people $p$ is at least

1. The problem is to find points on the graph of the function $$f(x) = \frac{1}{2}x^2 + \frac{3}{5}x + 7$$ and understand how to calculate the corresponding $y$ values for given $x

1. **State the problem:** Solve for $x$ in the equation $7.19 = 6.1 + \log\left(\frac{6}{0.03x}\right)$. 2. **Isolate the logarithm:** Subtract 6.1 from both sides to get

1. **State the problem:** Solve for $x$ in the equation $7.19 = 6.1 + \log\left(\frac{6}{0.3x}\right)$. 2. **Isolate the logarithm:** Subtract 6.1 from both sides:

1. **Problem statement:** A quadratic polynomial $p(x)$ has $(2x + 1)$ as a factor. When $p(x)$ is divided by $(x - 1)$ and $(x - 2)$, the remainders are $-6$ and $-5$ respectively