🧮 algebra

1. **State the problem:** Solve for $x$ in the equation $$\frac{e^{-x}}{e^x - e^x} = 2.$$\n\n2. **Analyze the denominator:** Notice that the denominator is $e^x - e^x$, which simpl

1. The problem appears to involve evaluating or simplifying an expression with variables and numbers: $b = 4, 7, 2, 9, 2, 3, 1, 2 + - (), c = 6, 2, 15, 7, 8, 1, 4 + \times d = 181,

1. The problem gives two functions: $f(x) = 3 - x^2$ and $g(x) = x + 1$. 2. We can analyze these functions individually or perform operations like addition, subtraction, multiplica

1. **Stating the problem:** Given two numbers $a = 5$ and $b = 7$, find the sum $a + b$. 2. **Formula used:** The sum of two numbers is given by the formula:

1. Énonçons le problème : résoudre une équation en utilisant la méthode de la balance consiste à effectuer la même opération des deux côtés de l'équation pour isoler la variable. 2

1. **State the problem:** Solve the inequality $2x^2 - 5x + 8 > 0$. 2. **Recall the quadratic inequality rule:** For a quadratic $ax^2 + bx + c > 0$, the sign depends on the parabo

1. **Stating the problem:** We are given multiple linear equations and asked to analyze their slopes and intercepts, and understand how the value of $y$ changes with respect to $x$

1. The problem asks to match each linear equation to the correct graphed line based on slope and y-intercept. (a) $y=\frac{2}{3}x+5$ has positive slope $\frac{2}{3}$ and y-intercep

1. The problem involves understanding and simplifying expressions with Greek letters and variables with subscripts, such as $\eta = \frac{\varepsilon}{x_\beta}$ and $\zeta = x_\bet

1. The problem involves understanding and simplifying algebraic expressions with Greek letters and variables. 2. We start with the given formulas and expressions such as $\eta = \f

1. **State the problem:** We are given the simultaneous equations: $$x - 6y = 10$$

1. The problem asks us to find the slope of a line using the formula $m = \frac{\text{rise}}{\text{run}}$. 2. The slope $m$ represents how steep the line is, calculated as the vert

1. **Problem 7:** Graph a horizontal line passing through $y=4$ with $x$ ranging from $-7$ to $7$. 2. **Formula and rules:** A horizontal line has the form $y = b$ where $b$ is a c

1. The problem is to evaluate the function $h(x) = 3x^x - 4$ at $x=3$. 2. The formula given is $h(x) = 3x^x - 4$. This means we first calculate $x^x$, then multiply by 3, and final

1. **State the problem:** Solve the inequality $$\frac{1}{3}(4-1) - \frac{1}{2}(x-3) \leq \frac{1}{4}$$ and illustrate the solution on a number line. 2. **Simplify the constants:**

1. **State the problem:** We need to determine which of the given quadratic expressions has a factor of $x+2b$, where $b$ is a positive constant. 2. **Recall the factor theorem:**

1. **State the problem:** Solve the equation $12c = 660$ for $c$. 2. **Formula and rules:** To solve for $c$, we use the rule of isolating the variable by dividing both sides of th

1. **State the problem:** David is eight times older than Nicole. David's age is 32 years. We need to find Nicole's age. 2. **Define variables:** Let $a$ be Nicole's age.

1. First, identify the function or equation you want to graph. This is usually given or derived from the problem. 2. Write the function in the form $y = f(x)$, where $y$ is the dep

1. The problem asks to identify the correct division equation from the given options. 2. Division equations relate a dividend, divisor, and quotient as: $$\text{dividend} \div \tex

1. The user provided four numbers in different formats: decimal, standard form, and scientific notation. 2. Let's identify each number: