🧮 algebra

1. **Problem statement:** Given the continuous function $f:\mathbb{R} \to \mathbb{R}$ satisfying $$f(x) - 2f(1-x) = 3x - 3 - e^{-x} + 2e^{x-1}$$

1. Stating the problem: Solve for $x$ in the equation $$478394(x+5) = 32678132468$$ and then evaluate the expression $$543728973482904 \div 3409244873 + 54378929 \times 438294 + 47

1. Masalah: Selesaikan persamaan $4x = 4$ untuk mencari nilai $x$. 2. Gunakan aturan dasar aljabar: untuk mengisolasi $x$, bagi kedua sisi persamaan dengan 4.

1. The problem asks to find the value of $4 \log 8 + 4 \log 32$. 2. Recall the logarithm property: $a \log b = \log b^a$.

1. **State the problem:** Simplify the expression $4 \log 8 + 4 \log 32$. 2. **Recall the logarithm property:** $a \log b = \log b^a$ and $\log x + \log y = \log (xy)$.

1. Diketahui persamaan logaritma: $\log_2(x) + \log_2(x - 2) = 5$. 2. Gunakan sifat logaritma bahwa $\log_a(m) + \log_a(n) = \log_a(m \cdot n)$, sehingga:

1. Diketahui persamaan \(\log_2(x) + \log_2(x - 2) = 5\). Kita diminta mencari nilai \(x\).\n\n2. Gunakan sifat logaritma: \(\log_a(m) + \log_a(n) = \log_a(m \times n)\). Jadi, \(\

1. **State the problem:** It takes 48 minutes for 7 people to paint 7 walls. We want to find how many minutes it takes 20 people to paint 20 walls. 2. **Understand the relationship

1. **State the problem:** It takes 48 minutes for 7 people to paint 7 walls. We want to find how many minutes it takes 20 people to paint 20 walls. 2. **Understand the relationship

1. **State the problem:** Solve the equation $ (4x-7)(x^2+2x) = 3 $ for $x$. 2. **Use the distributive property:** Expand the left side by multiplying each term in $4x-7$ by each t

1. **State the problem:** We need to compute the sum of the sequence: 3 + 6 + 9 + ... + 117 + 120 + 117 + ... + 9 + 6 + 3. 2. **Analyze the sequence:** This sequence increases by 3

1. The problem is to draw the graph of the function $f(x) = 4^x$. 2. The function $f(x) = a^x$ where $a > 0$ and $a \neq 1$ is an exponential function.

1. **State the problem:** Solve the linear equation $7x = 60 + 2x$ for $x$. 2. **Formula and rules:** To solve linear equations, we isolate the variable on one side by performing i

1. **State the problem:** Evaluate the expression inside the brackets and then write the function $y = x^3$ multiplied by that evaluated value. 2. **Evaluate the bracketed expressi

1. **Problem Statement:** Define even and odd functions, then find the domain and range of the function $f(x) = 1 + x^2$. 2. **Definitions:**

1. نبدأ بتحديد المشكلة: حل المعادلات الجبرية. 2. المعادلة هي تعبير رياضي يحتوي على مجهول (عادة x) وعلامة مساواة.

1. The problem is to understand the concept of a function in mathematics. 2. A function is a relation between a set of inputs and a set of possible outputs where each input is rela

1. Let's start by stating the problem: We want to understand how to solve a simple algebraic equation with the help of an example. 2. Consider the equation $2x + 3 = 7$. Our goal i

1. The problem: A magic square is a grid where the sums of numbers in each row, column, and diagonal are equal. 2. The formula: For an $n \times n$ magic square, the magic constant

1. **Problem:** Prove that for any natural number $k$, $1 \cdot k = k$. 2. **Formula and rules:** Multiplication by 1 is the identity operation in natural numbers, meaning multiply

1. **State the problem:** Prove by mathematical induction that $$\sum_{i=1}^n i^4 = \frac{n(n+1)(6n^3 + 9n^2 + n - 1)}{30}.$$\n\n2. **Base case (n=1):** Calculate the left side: $$