🧮 algebra

1. **State the problem:** Solve for $n$ in the equation $\left(32+\frac{1}{5}\right)-\left(7-\frac{5n}{3}\right)=\frac{5}{6}$.\n\n2. **Rewrite the equation:** \n$$32 + \frac{1}{5}

1. **State the problem:** We want to analyze the function $$y=4+\frac{1}{\sqrt{2-x}}$$ and understand its behavior. 2. **Identify the domain:** The expression under the square root

1. The problem states that the original recipe makes 2 1/2 cupcakes, but we need enough cupcakes for each person to have one, including herself. Let's assume the number of people i

1. **State the problem:** Simplify the expression $2xy + 45pv0 + 68cp8$. 2. **Analyze each term:**

1. Planteamos el problema: despejar la variable $x$ en la ecuación $$y=\frac{1}{\sqrt{2-x}}.$$\n\n2. Recordemos que para despejar $x$, primero debemos eliminar la raíz cuadrada y e

1. The problem is to calculate $4 \div \frac{5}{7}$ and express the answer as a fraction in its lowest terms. 2. Recall the rule for dividing fractions: dividing by a fraction is t

1. **State the problem:** We need to calculate $3 \div \frac{17}{5}$ and simplify the result. 2. **Recall the division rule for fractions:** Dividing by a fraction is the same as m

1. The problem is to multiply the fractions $\frac{9}{20}$ and $5$ and express the answer as a fraction in simplest form. 2. Recall the multiplication rule for fractions: $\frac{a}

1. **State the problem:** Calculate $\frac{9}{5} \times 4$ and express the answer as a fraction in its lowest terms. 2. **Write the multiplication:**

1. The problem is to understand the meaning or use of the variable $p$ in a mathematical context. 2. In algebra and many areas of mathematics, $p$ is often used as a variable or pa

1. Vamos resolver a inequação \(\log_2(x + 3) \leq 5 - \log_2(4 - x)\).\n\n2. Primeiro, isolamos os termos logarítmicos de um lado: \n\n\(\log_2(x + 3) + \log_2(4 - x) \leq 5\)\n\n

1. The problem asks for the reciprocal of the divisor given a numerator and denominator. 2. The reciprocal of a divisor is found by swapping the numerator and denominator.

1. **State the problem:** Simplify the expression $$\frac{x+1}{x-1} - \frac{1}{x} + x + \frac{1}{(x-1)^2}.$$\n\n2. **Find a common denominator:** The denominators are $x-1$, $x$, a

1. The problem asks to express the fractions $\frac{4}{5}$ and $\frac{1}{6}$ as equivalent fractions with denominator 30. 2. To find an equivalent fraction, multiply numerator and

1. **State the problem:** Find the 7th term of the geometric progression (GP) with first three terms 2, -10, 50, ... 2. **Recall the formula for the nth term of a GP:**

1. **State the problem:** We are given the first four terms of a geometric progression (GP): 1250, 250, 50, 10. We need to find:

1. **State the problem:** Calculate the value of $-45 - (-30)$. 2. **Recall the rule:** Subtracting a negative number is the same as adding its positive counterpart. So, $a - (-b)

1. **Problem statement:** We have three sequences and three expressions. We need to associate each sequence with its correct general term (term of order $n$). 2. **Sequences given:

1. Stwierdzenie problemu: Sprawdzimy, czy podane przekształcenia wyrażeń algebraicznych są poprawne. 2. Dane wyrażenia to:

1. **Problem:** Simplify the expression $$\frac{4}{3}$$. 2. **Formula and rules:** This is a simple fraction already in simplest form.

1. Simplify $\frac{8^8}{8^4}$. Use the rule $\frac{a^m}{a^n} = a^{m-n}$.