🧮 algebra

1. **Stating the problem:** We are given three pairs of linear equations and asked to analyze their relationships and solve the first pair.

1. **State the problem:** We want to find the set $W = \{x \in \mathbb{N} \mid 2x \leq 4 \wedge x > 1\}$. 2. **Understand the conditions:** The set $W$ contains natural numbers $x$

1. **State the problem:** Solve the equation $$\log \sqrt{3x + 1} = 1 + \log \sqrt{2x - 3}$$ for $x$. 2. **Recall logarithm properties:**

1. Planteamos el problema: La maquinaria pierde el 20% de su valor cada año, y su valor inicial es 40000. 2. La fórmula para el valor después de $n$ años con una pérdida porcentual

1. **State the problem:** Simplify the expression $$(3u^2 - 3u + 8) - (5u^2 - 4u - 1) + (2u^2 + 3u + 4)$$. 2. **Remove parentheses carefully:** Remember to distribute the minus sig

1. **State the problem:** Simplify the expression $$(-5t^2 - 4t + 1) + (-3t^2 + 7)$$. 2. **Formula and rules:** When adding polynomials, combine like terms. Like terms have the sam

1. **State the problem:** We are given two functions, $f(x)$ and $g(x)$, graphed as a blue decreasing line and a red upward-opening parabola, respectively. We need to find the poin

1. The problem asks which statement is true regarding the graphed functions $f(x)$ and $g(x)$. 2. We are given that $g(x)$ is the red increasing line in the center-right of the gra

1. **State the problem:** Solve the quadratic equation $$-2x^2 + 12x - 5 = 0$$ by finding the discriminant, describing the roots, and finding the exact solutions using the quadrati

1. **State the problem:** We are given the quadratic equation $x^2 + 2x + 7 = 0$. 2. **Find the discriminant:** The discriminant $\Delta$ of a quadratic equation $ax^2 + bx + c = 0

1. **Problem 1: Average height expression** The botanist has 4 plants with an average height of 23.2 cm. A fifth plant with height $x$ cm is added.

1. **Problemstellung:** Wir sollen die Gesamtkosten für die Herstellung eines Fußballs mit einem Durchmesser von 22 cm berechnen.

1. The problem asks us to describe the graph of the polynomial function $$p(x) = -2x^2 - x + 157$$ and identify its key features such as the direction it opens and its x-intercepts

1. **State the problem:** Solve the system of equations $$\begin{cases} -x + 4y = 6 \\ 2x - y = -5 \end{cases}$$

1. **State the problem:** We are given two functions $f(x) = x - 7$ and $g(x) = x^2 - 2x + 3$. We need to find the expression for the composition $g(f(x))$. 2. **Recall the composi

1. **State the problem:** Solve the system of equations: $$\begin{cases} 4x + 9y = 12 \\ -2x + y = -6 \end{cases}$$

1. **State the problem:** We have a cylindrical can with original radius $r = x$ cm and height $h = x$ cm.

1. **State the problem:** Determine if the point $(1,-2)$ satisfies the inequality $y \geq -6x - 3$. 2. **Substitute the point into the inequality:** Replace $x$ with $1$ and $y$ w

1. **State the problem:** We need to determine if the point $(3,4)$ satisfies the inequality $y < 4x - 2$. 2. **Write the inequality and substitute the point:** The inequality is $

1. **State the problem:** Solve the quadratic equation $$-\frac{\sqrt{7}}{2}x^2 - 7x - \sqrt{7} = 0$$. 2. **Identify coefficients:** The quadratic equation is in the form $$ax^2 +

1. **State the problem:** We are given two functions: \(f(x) = \left(\frac{3}{4}\right)^x\) and \(g(x)\) represented by a graph. We need to determine which statement about \(f(x)\)