🧮 algebra

1. **State the problem:** Solve the inequality $$3 \leq \frac{t}{2.52}$$ for $t$. 2. **Formula and rules:** To isolate $t$, multiply both sides of the inequality by 2.52. Since 2.5

1. **State the problem:** Solve the equation $9\ln(x+1) = -27$ for $x$. 2. **Recall the formula and rules:** The natural logarithm function $\ln(y)$ is the inverse of the exponenti

1. **State the problem:** Solve the system of equations: $$y=3x-2$$

1. **State the problem:** Solve the logarithmic equation $$\ln(2x - 8) - \ln(x - 4) = \ln(x)$$ for $x$. 2. **Recall the logarithm property:** The difference of logarithms can be wr

1. **State the problem:** Solve the system of equations and confirm the solution by graphing. 2. **General approach:** To solve a system of equations, we find values of variables t

1. **State the problem:** Write recursive and explicit equations for the given sequences. 2. **Part (a): Sequence 108, 120, 132, ...**

1. **State the problem:** Solve the equation $$\log_2(y^4) + \log_2(y) - \log_2(4y) = 2$$ for $y$. 2. **Recall logarithm properties:**

1. **State the problem:** Solve the equation $$2\ln(x) = \ln(e^5) - 2$$ for $x$. 2. **Recall properties of logarithms:**

1. **State the problem:** Simplify and solve the equation $$\frac{3x}{x^2 + 7x + 10} - \frac{x - 6}{x + 5} = - \frac{x^2 - 7x - 12}{(x + 2)(x + 5)}$$

1. The problem is to understand what happens to negative signs in algebraic expressions or equations. 2. Negative signs indicate the opposite or subtraction. When simplifying expre

1. **State the problem:** Simplify and evaluate the expression $$(-3)^4 + \frac{(-3)^6}{(-3)^5} - (-3)^2$$. 2. **Recall the rules:**

1. **Problem 1: Calculate the mean height of all 50 members of the basketball club.** 2. We have two groups:

1. Problem: Patricians and plutocrats are in the ratio 2:13, total 315. Find plutocrats. Formula: If ratio is $a:b$ and total is $T$, then part corresponding to $b$ is $\frac{b}{a+

1. Problem: The patricians and plutocrats are in the ratio 2 to 13, and the total number is 315. Find the number of plutocrats. 2. Formula: If two quantities are in ratio $a:b$ and

1. **State the problem:** We are given a table of values for $x$ and $y$: $$\begin{array}{c|ccccc}

1. Problem: The patricians and plutocrats are in the ratio 2 to 13, and the total number is 315. Find the number of plutocrats. 2. Formula: If two quantities are in ratio $a:b$ and

1. **State the problem:** Katrina claims that $3.5 \times 10^4$ is greater than $2.1 \times 10^6$ because $3.5 > 2.1$. 2. **Explain the mistake:** When comparing numbers in scienti

1. The problem asks to evaluate $$(9.27 \times 10^7) - (2.8 \times 10^4)$$ and express the result in scientific notation. 2. Recall that scientific notation expresses numbers as $a

1. **State the problem:** Simplify the expression $$\sqrt[3]{64 x^{28} y^{17} z^{4}}$$. 2. **Recall the cube root property:** The cube root of a product is the product of the cube

1. **State the problem:** We are given a graph of a linear function $F(x)$ and need to find its equation. 2. **Identify points from the graph:** The line passes through points appr

1. **State the problem:** Solve the inequality $$3 > \frac{11 - 5w}{9}$$ for $w$. 2. **Understand the inequality:** We want to isolate $w$ on one side. Since $w$ is inside a fracti