🧮 algebra

1. **Problem Statement:** We are given the function $f(x) = 2x - 3 \sin(x)$ and asked to analyze it. 2. **Formula and Explanation:** The function combines a linear term $2x$ and a

1. **Problem:** Solve the recurrence relation $$S(n) - 2S(n - 1) + S(n - 2) = 0$$ with initial conditions $$S(0) = 9$$ and $$S(1) = 10$$. 2. **Formula and approach:** This is a lin

1. **State the problem:** Find the value of $x$ such that $$\frac{x - 2}{x - 3} + \frac{x + 3}{x + 2} = 2.$$\n\n2. **Formula and rules:** To solve equations involving rational expr

1. **State the problem:** We have a line segment AB with points C and D between A and B. Car X travels \(\frac{3}{4}\) of AB from A to C, and Car Y travels \(\frac{1}{8}\) of BA fr

1. **State the problem:** Arrange the numbers 17%, \(\frac{3}{20}\), 0.14, and \(\frac{5}{36}\) in ascending order. 2. **Convert all numbers to decimals for easy comparison:**

1. The problem is to find the value of $x$ in the equation $$\frac{2x+3}{5} = 7.$$\n\n2. The formula used here is to solve linear equations by isolating the variable $x$.\n\n3. Fir

1. The problem asks for the exponential form of the product $108 \times 192$. 2. To express a number in exponential form, we first find its prime factorization.

1. The problem is to express the number $\frac{5}{2} \times 3.54$ as a product of prime factors with positive exponents. 2. First, calculate the product:

1. Stating the problem: Evaluate the expression $5(2 \cdot 3 \cdot 5^4)$.\n\n2. Formula and rules: We use the order of operations (PEMDAS/BODMAS) which tells us to calculate expone

1. The problem asks to draw the graph of $$y = -f(x)$$ given the graph of $$y = f(x)$$ with points (\-5,0), (\-3,\-3), (0,2), (1,3), (2,5). 2. The transformation $$y = -f(x)$$ refl

1. The problem states that the point $(1,4)$ lies on the graph of $f(x)$, meaning $f(1) = 4$. 2. We want to find which point lies on the graph of $y = f(x) + 3$.

1. **State the problem:** Simplify the expression $\sqrt{5(\sqrt{5}(\sqrt{5}))}$. 2. **Rewrite the expression:** The expression inside the square root is $5 \times (\sqrt{5} \times

1. **Problem statement:** Given the function $f(x) = \sqrt{x}$, find the functions for the following transformations: - Reflection across the x-axis

1. The problem is to verify or simplify the expression $$\sqrt{5} \sqrt[5]{5}$$ and compare it to $$5^{\frac{2}{7}}$$. 2. Recall the rules for exponents and radicals:

1. **State the problem:** Simplify or analyze the expression $180 - 2w^2 + 3w$. 2. **Identify the type of expression:** This is a quadratic expression in terms of $w$ because it co

1. Stwierdzenie problemu: Znajdź liczbę rozwiązań całkowitych równania $$\frac{13}{x} - \frac{19}{y} + \frac{11}{z} + \frac{247}{xy} - \frac{143}{xz} + \frac{209}{yz} = 1.$$\n\n2.

1. The problem is to understand the square root function and its properties. 2. The square root of a number $x$ is a value $y$ such that $$y^2 = x$$.

1. The problem is to evaluate the expression $$\frac{(5 + 1)^2}{(7 - 5)^2}$$. 2. The formula involves squaring sums and differences: $$(a + b)^2 = (a + b)(a + b)$$ and $$(a - b)^2

1. The problem asks to evaluate the squares of the expressions $(5 + 1)$ and $(7 - 5)$.\n\n2. Recall the formula for squaring a number: $a^2 = a \times a$.\n\n3. First, calculate $

1. The problem is to simplify the expression $cf - 1$. 2. Here, $c$ and $f$ are variables, and the expression represents the product of $c$ and $f$ minus 1.

1. **State the problem:** We are given two equations: