🧮 algebra

1. **Problem 11a:** Write inequalities for the ticket sales constraints. - Let $x$ be the number of adult tickets sold.

1. **Problem Statement:** Given the points $(-2,3)$, $(0,-1)$, $(1,1)$, and $(2,5)$, find the Lagrange interpolation polynomial $L(x)$ that passes through these points. 2. **Formul

1. Planteamos el problema: Tenemos que $3^x + 3^{-x} = \pi$ y $3^{2x} + 3^{-2x} = \pi^2$. Queremos encontrar el valor de $9^x + 9^{-x}$. 2. Observamos que $9^x = (3^2)^x = 3^{2x}$

1. Planteamos el problema: Dadas las ecuaciones $$3^x + 3^{-x} = \pi$$ y $$3^{2x} + e^{2(-x)} = \pi^2$$, hallar $$9^x + 9^{-x}$$. 2. Observamos que $$9^x = (3^2)^x = 3^{2x}$$ y $$9

1. **State the problem:** Solve the inequality $u - 4 > 16$ and simplify the answer. 2. **Formula and rules:** To solve inequalities, we isolate the variable on one side by perform

1. **State the problem:** Solve the inequality $$22 \leq x + 3$$ for $$x$$. 2. **Formula and rules:** To isolate $$x$$, subtract 3 from both sides of the inequality. When subtracti

1. **State the problem:** Solve for $x$ in the equation $$\frac{3^{2x-1}}{3^{x+1}} = 9.$$\n\n2. **Recall exponent rules:** When dividing powers with the same base, subtract the exp

1. **State the problem:** Solve for $x$ in the equation $$\frac{3^{2x-1}}{3^{x+1}} = 9.$$\n\n2. **Recall the properties of exponents:** When dividing powers with the same base, sub

1. **State the problem:** Solve the equation $4^x = \frac{32\sqrt{2}}{2}$ for $x$. 2. **Rewrite the right side:** Simplify the right side:

1. **State the problem:** Solve the equation $2^{2x-1} = \frac{1}{16}$ for $x$. 2. **Recall the formula and rules:** We know that $16 = 2^4$, so $\frac{1}{16} = 2^{-4}$ because $\f

1. Problema: Efectuar la operación $4 + \frac{1024}{128} - \frac{9}{3}$ paso a paso. 2. Fórmulas y reglas importantes:

1. The problem is to understand and analyze the given equation and the description of the parabola. 2. The equation appears to be a system or matrix-like expression: $$\begin{bmatr

1. Задача: Найти значение выражения $ (2.5 + 7)^2 $.\n\n2. Формула: Квадрат суммы двух чисел вычисляется по формуле $$ (a + b)^2 = a^2 + 2ab + b^2 $$.\n\n3. Подставим значения: $$

1. **Énoncé du problème :** Déterminer les réels $a$ et $b$ pour que la fonction $f(x) = x^2 + ax + b$ passe par les points $O(0;0)$ et $A(-1;5)$.

1. **State the problem:** Solve the linear equation $k + 2y = 4$ for $y$. 2. **Formula and rules:** To isolate $y$, we use the rule of solving linear equations by performing invers

1. **State the problem:** Solve the linear equation $2k + y = 4$ for $y$. 2. **Formula and rules:** To isolate $y$, subtract $2k$ from both sides of the equation.

1. **State the problem:** We need to find the amount of gas used by Denise's household given the unit charges for electricity, gas, and water as $x$, $y$, and $z$ respectively, and

1. **State the problem:** Solve for $x$ in the equation $3x - 5 = 16$. 2. **Write the formula and rules:** To solve for $x$, isolate $x$ by performing inverse operations. Addition

1. **State the problem:** We need to determine which tables represent a proportional relationship between $x$ and $y$. 2. **Recall the rule for proportional relationships:** Two va

1. **State the problem:** Solve for $n$ in the equation $-7 + n = 5$. 2. **Formula and rules:** To isolate $n$, add 7 to both sides of the equation to cancel out $-7$ on the left s

1. **State the problem:** Simplify the expression $$(8t - 5t^2) + 12(7t + 3)$$ and find the coefficient of $t$. 2. **Apply the distributive property:** Multiply 12 by each term ins