🧮 algebra

1. **State the problem:** Solve the linear equation $$\frac{1}{3}(x - 1) - \frac{1}{2}(1 - x + 2) = \frac{1}{4} \times \frac{1}{3} = \frac{1}{6}$$. 2. **Rewrite the equation clearl

1. **State the problem:** Solve the inequality $$\frac{1}{x} + \frac{1}{x+1} > \frac{3}{2}$$ for $x$. 2. **Find a common denominator and combine the left side:** The common denomin

1. **Planteamiento del problema:** Se tienen dos matrices cuadradas $A$ y $B$ de tamaño $2\times 2$: $$A = \begin{bmatrix} 3 & 5 \\ 2 & -6 \end{bmatrix}, \quad B = \begin{bmatrix}

1. **Planteamiento del problema:** Tenemos la función $$f(x)=\frac{x^{2}+x-2}{2x^{2}-2x-3}$$ y queremos graficarla y encontrar sus asíntotas. 2. **Factorización de numerador y deno

1. O problema é fatorar o polinômio quadrático $2x^2 + 7x - 4$. 2. Para fatorar um polinômio do tipo $ax^2 + bx + c$, usamos a técnica de encontrar dois números que multiplicados d

1. **Stating the problem:** We need to graph the function $$f(x) = \frac{x^2 + x - 2}{2x^2 - 2x - 3}$$ and analyze its key features. 2. **Formula and rules:** This is a rational fu

1. Planteamos el problema: Encontrar el vértice y los puntos de corte con los ejes de la parábola dada por la función $$y = -2x^2 - 2x + 4$$. 2. Fórmulas importantes:

1. O problema pede para resolver a equação $$\left|\frac{x+1}{x-1}\right|=2$$. 2. A definição do valor absoluto implica que $$\frac{x+1}{x-1}=2$$ ou $$\frac{x+1}{x-1}=-2$$.

1. **State the problem:** Divide the polynomial $x^3 + 2x^2 + 3x + 2$ by the binomial $x + 1$. 2. **Formula and rule:** Polynomial division can be done using long division or synth

1. **State the problem:** We are given a cubic polynomial $$x^3 + 4x^2 - kx - 2$$ and told that $$x - 1$$ is a factor of this polynomial. We need to find the value of $$k$$. 2. **R

1. The problem is to solve for $x$ in the equation $2x + 3 = 11$. 2. We use the basic algebraic principle: to isolate $x$, perform inverse operations to both sides of the equation.

1. Planteamos el problema: Simplificar y analizar la expresión $$t = (x-3)(x-5)(x+2)(x+4) - (x^2 - x - 13)^2 + 50$$. 2. Primero, expandimos los productos de dos binomios:

1. Planteamiento del problema: Simplificar completamente la expresión dada. 2. Recordemos algunas reglas importantes:

1. **State the problem:** Solve the inequality $5x + 3 < 2x - 1$. 2. **Write the inequality:**

1. **State the problem:** Solve the inequality $$\frac{x-1}{5} \geq x - \frac{2}{3}$$. 2. **Formula and rules:** To solve inequalities involving fractions, first eliminate denomina

1. The problem is to graph the line with slope 4 and y-intercept -2. 2. The formula for a line in slope-intercept form is $$y = mx + b$$ where $m$ is the slope and $b$ is the y-int

1. **State the problem:** Solve the equation $$1616x + 9 = 3^2 (2568^{3x - 4})$$ for $x$. 2. **Rewrite the equation:** The right side is $$3^2 (2568^{3x - 4}) = 9 \times 2568^{3x -

1. Planteamos el problema: calcular el valor de la expresión dada: $$(-2)^3 \left(\frac{1}{3}\right)^{-3} \left(-\frac{2}{3}\right) \sqrt[3]{\frac{-2 - \frac{1}{2} + \frac{1}{3}}{-

1. The problem is to solve the equation $$\frac{2x+3}{4} = 5$$ for $x$. 2. The formula used here is to isolate $x$ by eliminating the denominator and then solving the resulting lin

1. The problem is to solve the equation $$\frac{2x+3}{4} = 5$$ for $x$. 2. The formula used here is to isolate $x$ by eliminating the denominator and then solving the resulting lin

1. The problem is to solve the equation $$\frac{2x+3}{4} = 5$$ for $x$. 2. The formula used here is to isolate $x$ by eliminating the denominator and then solving the resulting lin