🧮 algebra

1. Problem: Calculate $4f(-1) + 3g(-2)$ given $f(x) = 3x - 7$ and $g(x) = 4x + 2$. 2. First, find $f(-1)$:

1. **Problem Statement:** (a) Find the domain of the function $$f(x) = \frac{x+3}{x^2 - 9}$$.

1. Stating the problem: Solve the inequality $$x^2 > 3x + 4$$. 2. Rearrange all terms to one side to compare to zero:

1. The given expression is a division chain: $7/4/3/2$. 2. According to the order of operations, division is performed from left to right.

1. We are asked to find the sum $$\sum_{i=0}^{n-1} i(i+1)$$ which means adding the values of $i(i+1)$ as $i$ goes from 0 to $n-1$. 2. Expand the term inside the summation:

1. **State the problem:** Simplify the expression $\frac{13}{25} + \frac{18}{5} - \frac{5}{10}$.\n\n2. **Find a common denominator:** The denominators are 25, 5, and 10. The least

1. The formulas for the sum of arithmetic series are typically denoted as $S_1 = \frac{n}{2}(a_1 + a_n)$ or $S_2 = \frac{n}{2}(2a_1 + (n-1)d)$. 2. Here, $n$ is the number of terms,

1. The problem is to simplify the expression $\frac{1}{4}x - \frac{3}{1}$. 2. Rewrite the terms clearly: $\frac{1}{4}x$ is one fourth of $x$, and $\frac{3}{1}$ is simply 3.

1. Simplify $\frac{\sqrt{24}}{\sqrt{12}}$. Step 1: Use the property $\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}$.

1. **Problem statement:** We have two linear equations: $$a \cdot x + b \cdot y = c$$

1. We need to find a closed form expression for the sum $$\sum_{i=0}^{n-1} i(i+1)$$. 2. First, expand the term inside the summation:

1. Écrivons chaque expression sous forme de puissance. 2. Pour A: $$A = \sqrt{7}^7 \times \sqrt{7}^{-13} = 7^{\frac{7}{2}} \times 7^{\frac{-13}{2}} = 7^{\frac{7}{2} + \frac{-13}{2}

1. Stating the problem: Solve the equation $$\frac{m - 3}{4} = \frac{m + 1}{3}$$ for $m$. 2. To eliminate the fractions, multiply both sides of the equation by the least common den

1. Estimate the sum of $\frac{2}{3} + \frac{6}{9}$. Simplify $\frac{6}{9} = \frac{2}{3}$. Sum: $\frac{2}{3} + \frac{2}{3} = \frac{4}{3} \approx 1.33$. 2. Estimate $\frac{3}{20} + \

1. The problem is to solve the equation $$x^x = 5^{x+25}$$ for $x$. 2. Rewrite the right side to separate the exponent, $$5^{x+25} = 5^x \cdot 5^{25}$$.

1. The problem is to evaluate the sum $$\sum_{i=1}^{n} i^2 (5-1)$$. 2. Simplify the term inside the sum: $$5-1 = 4$$, so the sum becomes $$\sum_{i=1}^{n} 4 i^2$$.

1. The first equation is $\frac{m - 3}{4} = \frac{m + 1}{3}$. We are given the LCD is 12. 2. Multiply both sides by 12 to clear the denominators:

1. The problem is to understand and evaluate \(\log_e\), which is the logarithm base \(e\) (known as the natural logarithm).\n\n2. The notation \(\log_e x\) is commonly written as

1. We are given the inequality $6 \leq -3(4-2x) < 8$ and need to solve for $x$. 2. Start by distributing $-3$ inside the parenthesis:

1. Enunciado del problema: Resolver la expresión $$\frac{\left(\frac{3}{1.2} + \sqrt{\frac{11}{25} + 1}\right) \div \left(-\frac{1}{2}\right) \cdot 3 \cdot \frac{2}{6}}{\frac{1}{5}

1. We are solving the equation: $$\frac{1}{2}(x + 6) = 4(x - 2)$$ 2. Start by distributing the terms on both sides: