🧮 algebra

1. **State the problem:** We are analyzing the expression $$\frac{2 - n}{\sqrt{n^2 + 3}}$$ and explaining why this expression is limited in terms of its domain and behavior. 2. **U

1. **State the problem:** Simplify the expression $$\frac{5}{6} \div 2 \frac{1}{3} + \frac{1}{2}$$. 2. **Convert mixed number to improper fraction:**

1. **Problem 1:** Find $x$ given that $|P| = -10$, where $$P = \begin{bmatrix} x+3 & x+2 \\ x+1 & x-1 \end{bmatrix}.$$ The determinant of $P$ is $$|P| = (x+3)(x-1) - (x+2)(x+1).$$

1. We are given the equation modeling the miles remaining to be paved after $x$ hours: $$y = 200 - 50x$$

1. The problem is to solve a quadratic equation generally expressed as $$ax^2 + bx + c = 0$$ where $a \neq 0$. 2. To solve it, we use the quadratic formula:

1. **State the problem:** Simplify the expression $4(5a + 3b) + 3(2a - b)$.\n\n2. **Distribute the coefficients:** Apply the distributive property by multiplying each term inside t

1. The problem asks to find the intersection $P \cap Q$ where: - $P$ is the set of prime factors of 210.

1. **State the problem:** Solve for $a$ given the equation $a^{\frac{1}{8}} = \sqrt{3}$.\n\n2. **Rewrite the equation:** Note that $\sqrt{3} = 3^{\frac{1}{2}}$. So the equation bec

1. **Énoncé du problème** : Montrer que $$\frac{1}{\sqrt{1}+\sqrt{2}} + \frac{1}{\sqrt{2}+\sqrt{3}} + \frac{1}{\sqrt{3}+\sqrt{4}} + \cdots + \frac{1}{\sqrt{99}+\sqrt{100}} = 9.$$\n

1. **Énoncé du problème :** Montrer que la somme

1. The problem asks for the change in temperature when the temperature drops from -2 to -4 degrees. 2. To find the change in temperature, subtract the final temperature from the in

1. Let's state the problem: We want to analyze the expression $$\frac{2n^2 - 3}{n^2 + 3}$$ and explain why this expression is limited (bounded). 2. First, observe the expression's

1. Problem statement: Jawad has two fish tanks, one larger and one smaller. The smaller tank initially has 35 fish.

1. State the problem: Find the number $x$ such that when it is added to 5, the result is the same as when $\frac{2}{3}$ of it is subtracted from 6. 2. Set up the equation based on

1. **State the problem:** Matt and Ming sold small and large boxes of oranges at a fundraiser. Matt sold 3 small and 14 large boxes for 203, and Ming sold 11 small and 11 large box

1. **State the problem:** We need to find the length and width of a rectangle given that its perimeter is 152 meters and the width is 22 meters less than the length. 2. **Define va

1. We are given the expression $$\frac{2^n}{3 - 2^{-n}}$$ and asked to find its smallest term. 2. First, rewrite the expression for clarity:

1. The problem asks us to find all factors of 40. 2. A factor of a number is any integer that divides that number exactly without leaving a remainder.

1. We start with 240 loaves of bread at the beginning of Monday. 2. On Monday, the bakery sold $\frac{1}{4}$ of the loaves, so the number of loaves sold is:

1. The problem is to find all multiples of 5 that are less than 51. 2. Multiples of 5 are numbers that can be written as $5n$, where $n$ is a positive integer.

1. **State the problem:** We need to solve the equation $$\frac{1}{3} - 4 = \frac{T}{5} + 1$$ for the variable $T$. 2. **Simplify the left side:** Convert $4$ into a fraction with