🧮 algebra

1. **State the problem:** Simplify the expression $\log_{2025} 2025 - \log_3 \left( \frac{1}{9} \right)$.\n\n2. **Recall the logarithm properties:**\n- $\log_a a = 1$ for any base

1. We start with the first equation: $3x^{2} - 4 = 1(-x - 2)$. 2. Distribute the right side: $3x^{2} - 4 = -x - 2$.

1. **State the problem:** Solve for $x$ in the equation $$4.0 \times 10^{-11} = x(2x + 0.10)^2.$$\n\n2. **Rewrite the equation:** Note that $0,10$ is a comma decimal notation, so r

1. **Stating the problem:** We have several tables and number lines showing relationships between quantities and their percent values. The goal is to find missing values marked as

1. **State the problem:** Solve the equation $2x^3 + 0.10x = 4.0 \times 10^{-11}$ for $x$. 2. **Rewrite the equation:**

1. **State the problem:** We need to find the cost of two types of concert tickets: a cheaper ticket and a more expensive ticket that costs 30 more than the cheaper one. A group bu

1. **Problem statement:** The perimeter of a rectangle is 16 cm and its area is 12 cm\textsuperscript{2}. Calculate the length and width of the rectangle. 2. **Formulas and rules:*

1. Il problema chiede di semplificare l'espressione $$\left(-\frac{9}{4} x^3 y^4 - \frac{1}{2} x^3 y^4\right) - \left(\frac{5}{8} x^3 y^4 - \frac{5}{4} x^3 y^4\right)$$. 2. Iniziam

1. **State the problem:** Factor the expression $$5y^2z - 42x^4y^5$$ completely. 2. **Identify common factors:** Look for the greatest common factor (GCF) in both terms.

1. **State the problem:** Factor completely the expression $$-46x^3 y z^4 - 29z$$. 2. **Identify the greatest common factor (GCF):** Look at each term:

1. **Problem:** Factor the quadratic expression $x^2 + 7x + 12$. 2. **Formula and rules:** To factor a quadratic $ax^2 + bx + c$ where $a=1$, find two numbers that multiply to $c$

1. The problem is to analyze the function $f(x) = x^2 - 7x + 12$. 2. This is a quadratic function of the form $f(x) = ax^2 + bx + c$ where $a=1$, $b=-7$, and $c=12$.

1. **State the problem:** Solve the equation $$\frac{x}{2}(x+5) - \frac{1}{3}(x-2) = 0$$ for $x$. 2. **Write the equation clearly:** $$\frac{x}{2}(x+5) - \frac{1}{3}(x-2) = 0$$

1. **Problem:** Solve the multistep equation $3(2x - 6) + x = 3$ for $x$. 2. **Formula and rules:** Use the distributive property and combine like terms.

1. The problem states the relationship between the number of ravioli $r$ and the number of people $y$ as $y = \frac{1}{4}r$. 2. This means for every 4 ravioli, there is 1 person. T

1. **Problem statement:** A zoo keeper wants to fence a rectangular habitat for goats. The length $L$ should be at least 80 feet, and the perimeter $P$ should be no more than 300 f

1. **State the problem:** We are given inequalities involving $x$ and $y$:

1. Stating the problem: Solve for $x$ in the equation $$\sqrt{x} + 4! + \sqrt{x} = 12$$. 2. Recall that $4!$ (4 factorial) means $4 \times 3 \times 2 \times 1 = 24$.

1. **State the problem:** Solve the inequality $$|x + 6| \leq 8$$ algebraically. 2. **Recall the rule for absolute value inequalities:** For any real number $A$ and positive number

1. **State the problem:** Solve the inequality $$|x - 6| < 7$$ algebraically. 2. **Recall the rule for absolute value inequalities:** For any real number $a$ and positive number $b

1. The problem is to understand and express the number 9.2 × 10^{-11}. 2. This is a number written in scientific notation, where 9.2 is the coefficient and 10^{-11} means we multip