🧮 algebra

1. **נתון:** הפונקציה $$f(x)=\frac{e^{2x} - 9e^x}{e^{2x} - 10e^x + 9}$$. 2. **מציאת תחום ההגדרה:**

1. נתחיל בהגדרה כללית: משוואה היא ביטוי מתמטי שמכיל שני צדדים שמחוברים באמצעות סימן שוויון $=$. 2. מטרת פתרון המשוואה היא למצוא את הערך או הערכים של המשתנה שמקיימים את השוויון.

1. **State the problem:** Solve the equation $$9^x + 9 = 10 \cdot 3^x$$ for $x$. 2. **Rewrite the bases:** Note that $9 = 3^2$, so rewrite $9^x$ as $(3^2)^x = 3^{2x}$.

1. **נתונה הפונקציה:** $$f(x) = \frac{e^{2x} - 9e^x}{e^{2x} - 10e^x + 9}$$

1. Solve $5^{-x} \cdot 5^{x-2} = \frac{25^{2x}}{5}$ Combine powers on the left: $5^{-x + x - 2} = 5^{-2}$

1. **State the problem:** We are given the function $$f(x) = \frac{e^{2x} - 9e^x}{e^{2x} - 10e^x + 9}$$ and want to analyze its behavior. 2. **Simplify the expression:** Let $$y =

1. Problem 10: Given $f(x)= \frac{x}{3x+2}$ and $g(x) = -\frac{4}{x}$, find $g \circ f$ and its domain. 2. To find $g(f(x))$, substitute $f(x)$ into $g$:

1. The problem is to factor the quadratic expression $x^2 + 5x + 6$. 2. We look for two numbers that multiply to the constant term 6 and add up to the coefficient of $x$, which is

1. **Problem:** If the roots of the equation $ax^2 + cx + c = 0$ are in the ratio $p : q$, show that $$\sqrt{\frac{p}{q}} + \sqrt{\frac{q}{p}} + \sqrt{\frac{c}{a}} = 0.$$ Step 1: L

1. Stating the problem: Solve for $k$ in the equation $$\frac{-13.3 + k}{11.796} = -0.296.$$ 2. Multiply both sides by 11.796 to isolate the numerator:

1. The problem states that $\sqrt[3]{\sqrt{x}} = 2$. We need to find the value of $x$ that satisfies this equation. 2. Rewrite the equation using exponents: $\sqrt{x} = x^{\frac{1}

1. **State the problem:** Solve for $x$ in the equation $$\frac{2}{2x^2 + 3x - 1} = \frac{1}{x^2 - 1}.$$\n\n2. **Rewrite the equation:** Cross-multiply to eliminate the denominator

1. **State the problem:** Solve the equation $$(x - 3)^{\frac{3}{5}} = 8$$ for $x$. 2. **Isolate the expression:** The equation is already isolated with the power expression on one

1. The problem is to check the value of an expression or function at $x = -8.5$. 2. Since the user did not specify the exact expression or function, let's assume a general example:

1. The problem is to solve the equation and then check the solution for the number 2. 2. Since the user did not specify the exact equation, let's assume the equation is $x = 2$.

1. The problem is to find the slopes of the linear equations numbered 12 to 22. 2. Recall that the slope of a line in the form $Ax + By = C$ is given by $m = -\frac{A}{B}$.

1) Rendre le dénominateur entier naturel : A = \frac{\sqrt{7} - \sqrt{3}}{\sqrt{7} + \sqrt{3}}

1. The problem is to find the slopes from given linear equations. 2. Recall that the slope of a line in the form $y = mx + b$ is the coefficient $m$ of $x$.

1. The problem is to find the slope of a line given its equation. 2. The slope of a line in the form $y = mx + b$ is the coefficient $m$ of $x$.

1. The problem is to rewrite and analyze the given linear equations from 11 to 18. 2. For equation 11: Given \(3x + 2y = 6\) and \(2y = -3x + 6\), both represent the same line. Sol

1. **Problem 1:** Simplify and verify the expression $$\frac{2}{x+2} + \frac{4}{x-2} = \frac{x-1}{x^2-4}$$. 2. First, note that $$x^2 - 4 = (x+2)(x-2)$$.