🧮 algebra

1. **Problem Statement:** An element with an initial mass of 430 grams decays by 27.4% per minute. We want to find how much of the element remains after 19 minutes, rounded to the

1. **State the problem:** Solve the equation $$2x^2 + xy - y^2 = 2$$ for $y$ in terms of $x$. 2. **Identify the type of equation:** This is a quadratic equation in $y$ with terms i

1. **Problem:** Solve the proportion $$\frac{2x + 3}{2} = \frac{6x - 3}{5}$$ for $x$. 2. **Formula and rules:** To solve proportions, cross-multiply:

1. **State the problem:** Find the inverse function of $g(x) = x^3$. 2. **Recall the definition of inverse function:** The inverse function $g^{-1}(x)$ satisfies $g(g^{-1}(x)) = x$

1. **Énoncé du problème** : Résoudre dans $\mathbb{R}$ l'équation $\frac{3x+1}{x} = \frac{2 - x}{2}$. 2. **Formule et règles importantes** : Pour résoudre une équation de fractions

1. **State the problem:** Find the inverse function of $h(x) = x^2 - 2x + 1$. 2. **Rewrite the function:** Notice that $h(x) = x^2 - 2x + 1$ can be rewritten by completing the squa

1. **Énoncé du problème :** Résoudre dans $\mathbb{R}$ l'équation $$\frac{3x+1}{x} = \frac{2-x}{2}.$$ 2. **Formule et règles importantes :** Pour résoudre une équation de fractions

1. **State the problem:** Find the inverse function of $f(x) = 2x + 1$. 2. **Recall the formula and rules:** The inverse function $f^{-1}(x)$ reverses the effect of $f(x)$. To find

1. The problem is to solve the equation $$2^x = x^2$$ for the value(s) of $x$. 2. This is a transcendental equation where an exponential function equals a polynomial function. Such

1. **State the problem:** Compare the two fractions $\frac{111}{1111}$ and $\frac{1111}{11111}$ to determine which is greater. 2. **Formula and approach:** To compare two fractions

1. **Problem 1:** Calculate $a + b + c + d + e + f$ if $abcdef = 6(defabc)$. 2. We are given the equation:

1. The problem asks to find which fraction is equivalent to the mixed number $2 \frac{3}{4}$. 2. First, convert the mixed number to an improper fraction.

1. نبدأ بحل التمرين الأول: بسط الأعداد التالية ثم نذكر أصغر مجموعة ينتمي إليها كل عدد. 2. العدد الأول: $$\frac{2}{\sqrt{3}+1}$$

1. **State the problem:** Find the value of $x$ that satisfies the equation $$\frac{2x-3}{5} = 7.$$\n\n2. **Formula and rules:** To solve for $x$ in a linear equation with a fracti

1. **Problem Statement:** We are given the function $f(x) = \sqrt[5]{x}$ and its transformations: - $f(x) = \sqrt[5]{x} + 6$

1. **Problem Statement:** Simplify the expression $$\frac{\left[4^{5} \cdot (64)^{3} \cdot 2^{3}\right]^{\frac{1}{2}}}{8^{5} \cdot (128)^{2}}$$

1. Let's start by stating the problem: A mixed inequality involves two inequalities joined together, such as $a < x \leq b$. It means $x$ is a number that satisfies both inequaliti

1. **State the problem:** Solve the equation $\frac{1}{2}(4x+8)=10$ for $x$. 2. **Use the distributive property:** Multiply both sides by 2 to eliminate the fraction:

1. **State the problem:** Solve the equation $9 - x = 3x + 1$ for $x$. 2. **Write down the equation:**

1. The problem asks to write the given numbers as non-terminating decimal fractions (كسور عشرية غير زائدة). 2. To convert a number to a decimal fraction, we express it as a fractio

1. **State the problem:** Simplify the expression $\sqrt{10} + 4\sqrt{10}$. 2. **Recall the rule:** When adding terms with square roots, you can only combine them if the radicands