🧮 algebra

1. **Problem statement:** Given the function $$y = x^3 + x^2 - 4x - 4$$, answer the following: 2. **End behavior:** To find the end behavior, analyze the leading term of the polyno

1. **Problem Statement:** Describe the transformations of the graph for the function $$y = -(x - 2)^2 - 4$$. 2. **General form and rules:** The parent function is $$y = x^2$$.

1. The problem is to analyze the piecewise function $$y=\begin{cases} x & \text{if } x<0 \\ x^2 & \text{if } x\geq 0 \end{cases}$$. 2. This function is defined differently for nega

1. **Problem statement:** Evaluate the expression $7 \frac{3}{8} + 4 \frac{1}{8}$ using number lines and rational number addition. 2. **Formula and rules:** To add mixed numbers, f

1. **State the problem:** Solve the exponential equation $$9^{x+1} = 243^{x+1}$$ for all values of $x$. 2. **Rewrite bases as powers of the same base:**

1. **State the problem:** Solve the equation $5(3x-4)+11=12x$ for $x$. 2. **Apply the distributive property:** Multiply 5 by each term inside the parentheses.

1. **State the problem:** Solve the equation $7x - 1 = 4x + 8$ for $x$. 2. **Write down the equation:**

1. **State the problem:** Solve for $y$ in the equation $\frac{4.5}{y} = \frac{12.5}{4}$.\n\n2. **Write the formula and explain:** This is a proportion equation where two ratios ar

1. **State the problem:** Solve the equation $$27y - 3 = 3(9y - 1)$$ for $y$. 2. **Apply the distributive property:** Expand the right side:

1. **State the problem:** Solve the equation $$6 - 2(a + 1) = 7 + a$$ for the variable $a$. 2. **Apply the distributive property:** Multiply $-2$ by each term inside the parenthese

1. Énonçons le problème : Résoudre l'inéquation $$\frac{1}{x} + \frac{2}{x^2} + \frac{1}{x^3} \geq 0$$. 2. Pour résoudre cette inéquation, mettons tous les termes sous un dénominat

1. **State the problem:** We have an exponential function $f(x) = ab^x$ and two points: $f(-3.5) = 22$ and $f(2) = 64$. We need to find $f(1)$ to the nearest tenth. 2. **Write the

1. Énonçons le problème : Résoudre l'inéquation $$\frac{1}{x} \leq 5$$. 2. Rappelons la règle importante : pour résoudre une inéquation impliquant une fraction, il faut considérer

1. **State the problem:** Find the equation of the line passing through the points $(-12, -6)$ and $(0, 6)$ in slope-intercept form $y=mx+b$. 2. **Formula for slope:** The slope $m

1. **State the problem:** Simplify the fraction $\frac{1025}{200}$. 2. **Formula and rules:** To simplify a fraction, divide numerator and denominator by their greatest common divi

1. **State the problem:** Solve the equation $$x^4 - 18x^2 + 81 = 0$$ to find the solution set. 2. **Rewrite the equation:** Notice that the equation is quadratic in form if we let

1. **Énoncé du problème :** Calculer et simplifier l'expression $$\frac{\sqrt{5} - \sqrt{2}}{\sqrt{5} + \sqrt{2}} - \frac{\sqrt{5} + \sqrt{2}}{\sqrt{5} - \sqrt{2}}.$$\n\n2. **Formu

1. **State the problem:** Solve the system of equations: $$\begin{cases} x + y = 13 \\ x - 2y = 5 \end{cases}$$

1. **State the problem:** Find the value of $a_2$ if $\frac{2}{6} - 8$. 2. **Simplify the expression:**

1. **State the problem:** A population increases by 5% each year. We need to find the total percentage increase after two years. 2. **Formula used:** When a quantity increases by a

1. المشكلة: تبسيط وتحقق من صحة المعادلة $M = (2x - 5)(4x + 3) = 4x^2 - 20x + 23$. 2. نستخدم خاصية التوزيع (توزيع الحد الأول على الثاني):