🧮 algebra

1. **State the problem:** Simplify the expression $$A = \frac{3x^2 - 12x + 12}{x^2 - 5x + 6} \cdot \frac{6x^3 - 54x}{x^3 - 6x^2 + 9x} - \frac{x^2 + x - 6}{2(x^2 - 9)}$$

1. El problema nos da una función por partes: $$f(x) = \begin{cases} -x + 2 & \text{si } x < -1 \\ 1 & \text{si } -1 \leq x \leq 1 \\ x + 2 & \text{si } x > 1 \end{cases}$$

1. **State the problem:** We need to determine whether each given exponential function represents exponential growth or exponential decay. 2. **Recall the rule:** For an exponentia

1. Planteamos el problema: expresar la función $$y = \frac{x^2}{\sqrt{1 - x^2}}$$ en función de $$r$$. 2. Para hacer esto, necesitamos una relación entre $$x$$ y $$r$$. Sin embargo

1. **State the problem:** Simplify the expression $64^{\frac{2}{3}}$. 2. **Recall the rule for fractional exponents:** For any positive number $a$ and rational exponent $\frac{m}{n

1. The problem asks to write the inequality represented by the number line. 2. The number line shows numbers 3, 4, 5, 6, 7, 8 with a blue dot on 5 and an arrow pointing right from

1. **State the problem:** Factor the quadratic expression $x^2 - 10x + 25$. 2. **Recall the factoring formula:** For a quadratic expression $ax^2 + bx + c$, if it can be factored a

1. **State the problem:** We are given a number line with an open circle at 4 and an arrow pointing to the right from 4. 2. **Interpret the open circle:** An open circle at 4 means

1. **State the problem:** Simplify the quadratic expression $x^2 - 10x + 25$. 2. **Recall the formula:** A perfect square trinomial has the form $a^2 - 2ab + b^2 = (a - b)^2$.

1. The problem is to understand the relationship between fractions and division. 2. A fraction \( \frac{a}{b} \) represents the division of \( a \) by \( b \), where \( a \) is the

1. **State the problem:** Two friends, Hana and Freya, have different bills for printing photographs. Hana pays 16.50 for 150 photos, and Freya pays 9.60 for 80 photos. We need to

1. The problem asks to convert the logarithmic equation $\log_3 27 = 3$ into its equivalent exponential form. 2. Recall the definition of logarithm: $\log_b a = c$ means $b^c = a$.

1. El problema es resolver una inecuación, que es una desigualdad matemática. 2. La regla principal para resolver inecuaciones es que al multiplicar o dividir por un número negativ

1. **State the problem:** We want to understand how to solve or analyze the function $y=(x-1)^2 + 2$ on a graph. 2. **Formula and explanation:** This is a quadratic function in ver

1. **State the problem:** Simplify the expression $-2x^2(4x - x^2 - 3)$. 2. **Use the distributive property:** Multiply $-2x^2$ by each term inside the parentheses.

1. **State the problem:** We need to determine who has the better deal for printing photographs between Hana and Freya. 2. **Understand the concept:** The cost is directly proporti

1. **Problem 13:** Given the piecewise function $$g(x) = \begin{cases} ax + 3, & x \leq 1 \\ (x + a)^2 - 10, & x > 1 \end{cases}$$

1. The problem states that Gwen starts with 10 spools of thread and uses 2 spools per dress sewn. 2. The relationship between dresses sewn ($d$) and spools remaining ($s$) is linea

1. The problem asks: With 6 seed packets, how many total plants can Rebecca have in her backyard? 2. From the graph description, the relationship between seed packets and plants is

1. El problema nos pide encontrar la función inversa $f^{-1}(x)$ de la función dada $f(x) = 2x - 13$. 2. Para encontrar la función inversa, recordemos que si $y = f(x)$, entonces l

1. El problema nos pide determinar el rango de la función $$f(x) = x^2 - 4x - 9$$ con dominio $$x \in (-3,3)$$. 2. Para encontrar el rango, primero identificamos que $$f(x)$$ es un