🧮 algebra

1. **State the problem:** We need to find the horizontal and vertical asymptotes, new reference points, domain, and range of the function $$f(x) = \frac{2x^2 - 4x}{x^2 + 4x + 4}$$

1. The problem is to analyze the function $g(x) = -8(3)^x$. 2. This is an exponential function of the form $g(x) = a \cdot b^x$ where $a = -8$ and $b = 3$.

1. **State the problem:** We are given the function $f(x) = \frac{3}{5} 4^x$ and need to write the limit statements for its left and right end behaviors. 2. **Recall the behavior o

1. **State the problem:** Simplify or analyze the expression $$\frac{5}{2} x^6 - \frac{1}{2} x^4$$. 2. **Identify the terms:** The expression has two terms: $$\frac{5}{2} x^6$$ and

1. **State the problem:** Simplify the expression $9m^7 - 3m^2$. 2. **Identify common factors:** Both terms have a common factor of $3m^2$.

1. **State the problem:** Solve the system of linear equations: $$\begin{cases} x + 2y = 5 \\ 5x + 6y = 9 \end{cases}$$

1. **State the problem:** Solve the system of equations using Gauss-Jordan elimination: $$\begin{cases} x + y - z = -1 \\ -x + 5y + 19z = 9 \\ -5x + 5y + 3z = 8 \end{cases}$$

1. **State the problem:** Solve the system of linear equations using Gauss-Jordan elimination: $$\begin{cases}-0.3x - 0.8y = -9 \\ 0.6x - 1.4y = -15 \end{cases}$$

1. **State the problem:** We need to find how many adults and children attended the movie given that 61 people attended in total and the total amount paid was 472. 2. **Define vari

1. **State the problem:** A street vendor has 360 T-shirts total, some short sleeve and some long sleeve. Short sleeve shirts sell for 14 each, long sleeve for 16 each. Total sales

1. The problem asks to evaluate the expression $2x - y$ given the values $x=9$ and $y=-14$. 2. The formula is simply the expression itself: $$2x - y$$

1. **State the problem:** Simplify the expression $$-2x^4y\sqrt{16y^{13}} - 2^{3}\sqrt{16x^{12}y^{16}} - x^{2}y^{3}\sqrt[3]{-54x^{6}y^{7}}$$ 2. **Recall the rules:**

1. **State the problem:** Solve the equation $$-0.8(10 - x) = 36$$ for $x$. 2. **Use the distributive property:** Multiply $-0.8$ by each term inside the parentheses.

1. **State the problem:** Expand and simplify the expression $\frac{1}{2}(T - 4)^2$. 2. **Recall the formula:** The square of a binomial $(a - b)^2$ is expanded as $$ (a - b)^2 = a

1. **State the problem:** Solve the equation $x^2 + \frac{3}{2}x = 0$. 2. **Formula and rules:** To solve quadratic equations, we can factorize or use the quadratic formula. Here,

1. **State the problem:** Solve the equation $$\frac{x}{9} = \frac{5}{x}$$ for $x$. 2. **Use the cross-multiplication rule:** When two fractions are equal, their cross products are

1. State the problem. Solve for $w$ in the equation $5w + 9 = 39$.

1. The problem is to graph the linear inequality $y \geq 10 - x$. 2. First, we graph the boundary line $y = 10 - x$. This is a straight line with slope $-1$ and y-intercept $10$.

1. Problem: Kürzen Sie den Bruchterm $\frac{3ab}{6a}$ soweit wie möglich. 2. Formel: Um Brüche zu kürzen, teilt man Zähler und Nenner durch ihren größten gemeinsamen Teiler (ggT).

1. **State the problem:** Solve the factoring problem $$3x^2 - 18x = -27$$. 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero:

1. The problem asks to identify the vertex, focus, directrix, and axis of symmetry for the parabola given by the equation $$y = -3$$ and related information. 2. Since $$y = -3$$ is