🧮 algebra

1. **State the problem:** We analyze the graph of a rational function to find its domain, range, intercepts, and asymptotes. 2. **Domain:** The domain is all real numbers except wh

1. **Problem Statement:** We analyze a hyperbola-like rational function with two branches, having vertical asymptote at $x=1$ and horizontal asymptote at $y=2$.

1. **State the problem:** We analyze the function's domain, range, intercepts, and asymptotes based on the description of a hyperbola-like rational function with two branches.

1. Das Problem lautet: Wie lange dauert es, bis das nächste Level erreicht wird, wenn die aktuelle Erfahrung (XP) 1.44 \times 10^{304} von 1.65 \times 10^{304} XP ist und die XP-Ra

1. **State the problem:** Write an equation of a parabola with a vertex at the origin and a directrix at $y=5$. 2. **Recall the definition and formula:** A parabola is the set of p

1. **State the problem:** Factor the cubic expression $$2x^3 - 13x - 24$$. 2. **Recall the factoring approach:** For cubic polynomials, try to find rational roots using the Rationa

1. **State the problem:** Sam plans to order a total of 144 items. We need to find how many helmets will be in this new order. 2. **Identify the quantities and relationships:** The

1. **State the problem:** Factor the quadratic expression $$5x^2 + 19x + 18$$. 2. **Recall the factoring formula:** For a quadratic $$ax^2 + bx + c$$, we look for two binomials $$(

1. **State the problem:** Find the value of $p^2$ when $p=4$. 2. **Formula used:** The square of a number $p$ is given by $$p^2 = p \times p$$

1. **State the problem:** Find the value of $q^2$ when $q=6$. 2. **Formula used:** The square of a number $q$ is given by $$q^2 = q \times q$$.

1. **State the problem:** Subtract the rational expressions $$\frac{2b - 12y}{7b} - \frac{6b + 5y}{7b}$$ and simplify the result. 2. **Formula and rules:** When subtracting rationa

1. **State the problem:** Simplify the expression $2(5a+8a)-3(4a-8)$. 2. **Apply the distributive property:** Multiply each term inside the parentheses by the factor outside.

1. The problem is to solve the equation $2x + 3 = 11$ for $x$. 2. The formula used here is to isolate $x$ by performing inverse operations. We subtract 3 from both sides and then d

1. **Énoncé du problème :** Remplir un carré magique multiplicatif 3×3 avec des puissances de $a$ et $b$ données, de sorte que le produit de chaque ligne, colonne et diagonale soit

1. **State the problem:** We need to rewrite the quadratic function $$g(x) = 5x^2 + 20x - 12$$ in the form $$g(x) = a(x + h)^2 + k$$ where $$a, h, k \in \mathbb{Z}$$. 2. **Recall t

1. **State the problem:** Multiply and divide the rational expressions: $$\frac{4x^3+4x^2}{2x} \div \frac{x^2-1}{x^2+2x-3} \cdot \frac{x+3}{x^2+3x}$$

1. **State the problem:** Multiply and divide the rational expressions: $$\frac{x-1}{x^2+5x+6} \cdot \frac{x^2+x-2}{3} \div \frac{x-1}{3x}$$

1. **State the problem:** Solve the equation $$-\frac{1}{2} = 3 - \left(\frac{1}{2}x - 10\right)$$ for $x$ and write the answer in simplest form. 2. **Rewrite the equation:**

1. **Problem 9:** Find the equation of an equilateral hyperbola with foci at $(\sqrt{2},0)$ and $(-\sqrt{2},0)$. - An equilateral hyperbola has equal transverse and conjugate axes,

1. **State the problem:** Solve the inequality $$w + 3 \leq -10$$ and determine if $$w = -13$$ is a solution to the inequality. 2. **Rewrite the inequality:** To isolate $$w$$, sub

1. **State the problem:** Solve the inequality $w + 3 \leq -10$. 2. **Find the boundary point by solving the equation:**