🧮 algebra

1. **State the problem:** Solve the equation $$\log_{10}(x - 3) + \log_{10} x = \log_{10} 12$$ for $x$. 2. **Apply logarithm properties:** Recall that the sum of logarithms with th

1. State the problem: Solve the equation $$\frac{3}{2}x + 3 - \frac{1}{2}x + 1 = \frac{1}{x} + 1$$ for $$x$$. 2. Simplify the left side by combining like terms:

1. Stated problem: Solve the equation $$3 \log_{10}{(x-3)} + \log_{10}{x} = \log_{10}{12}$$. 2. Use logarithm property: $$a\log_b{c} = \log_b{c^a}$$, so rewrite the equation as $$\

1. نبدأ بتعريف الدالة $g(x)$، حيث نعلم أن $g(-2)=a$ لبعض القيمة $a$. 2. المطلوب هو حساب قيمة الدالة المركبة $(g \, o \, g \,...\, o g)(-2)$ حيث تم تطبيق $g$ على نفسها $n$ مرات متتا

1. **State the problem:** Find the value of $x$ from the equation $\log_2 x + \log_2 (x-3)$.\n\n2. **Use logarithm properties:** Recall that $\log_a m + \log_a n = \log_a (m \times

1. **Énoncé du problème :** Montrer que

1. Given the equation $x \log_2 x + \log_2 (x - 3) = 2$, we need to find the value of $x$. 2. Recall that $\log_2 (x - 3)$ is defined only if $x - 3 > 0$, so $x > 3$.

1. State the problem: Determine which of the given choices is not in the interval $-\tfrac{1}{2} < x < \tfrac{2}{3}$. 2. Interpretation and method: A number is in the interval if i

1. The problem asks for a pair of factors of -90 whose sum is -1. 2. We start by listing pairs of numbers that multiply to -90.

1. The problem is to determine the relationship between the fractions $\frac{-1}{2}$ and $\frac{2}{3}$ in the context of set inclusion, denoted by $\subseteq$. 2. Set inclusion $A

1. The problem is to find the best approximation interval for $\sqrt{91}$.\n\n2. We know that $9^2 = 81$ and $10^2 = 100$. Since $91$ is between $81$ and $100$, its square root mus

1. Stating the problem: Simplify the expression $$3(6 - x) / (x^3 + 27) + (x+3) / (x+4) - (x-3) / (x^2 - 3x + 9) - (x-4) / (x-3) - 1 / (x+3) + 7 / (x^2 + 3x - 4)$$

1. The problem asks for the best approximation of $\sqrt{91}$.\n\n2. We note the perfect squares near 91: $81 = 9^2$ and $100 = 10^2$.\n\n3. Since $91$ is between $81$ and $100$, i

1. The problem asks to solve an equation or expression using decomposition, but the specific problem is missing. 2. Decomposition generally means breaking down a complex expression

1. **State the problem:** Simplify the expression $$\frac{3(6-x)}{x^3+27} + \frac{x-3}{x^2-3x+9} - \frac{1}{x+3}.$$\n\n2. **Factor denominators where possible:**\n- $x^3+27$ is a s

1. **Problem Statement:** Factor the cubic polynomial $$6x^3 + 33x^2 + 45x$$. 2. **Identify the greatest common factor (GCF):** The coefficients 6, 33, and 45 share a common factor

1. **State the problem:** Find the factors of 324 that add up to 15. 2. **Find the factors of 324:**

1. Let's restate the problem: Find a factor of 300 such that when the factor and some other number are added, the sum equals -65. 2. First, list the factors of 300. Since 300 = $2^

1. **State the problem:** Determine on which intervals the function $f(x) = |9 - x^2|$ is decreasing. 2. **Analyze the function:** Inside the absolute value, we have the quadratic

1. **Stating the problem:** We have a cubic function $$y = ax^3 + bx^2 + cx + d$$ that passes through points $(3,0)$ and $(0,6)$, and has a critical point at $(2,2)$. 2. **Using gi

1. Kita diminta menentukan banyaknya bilangan bulat $a$ dengan $0 \leq a \leq 10$ sehingga $$\frac{a(a^2-1)}{3}$$ merupakan bilangan asli. 2. Bentuk tersebut dapat ditulis ulang se