🧮 algebra

1. **State the problem:** Solve for $x$ in the equation $$\frac{x}{2} - \frac{3 - x}{4} = 0.$$\n\n2. **Formula and rules:** To solve equations with fractions, find a common denomin

1. The problem asks to find the monthly tax due on an income of P60,000. 2. To solve this, we need the tax rate or tax formula, which is not provided in the question. Assuming a ta

1. The problem is to find the value of $\ln(4)$, where $\ln$ denotes the natural logarithm, which is the logarithm to the base $e$ (Euler's number, approximately 2.71828). 2. The n

1. **State the problem:** Solve the exponential equation $$e^{2x} - 5e^{x} + 4 = 0$$ for $x$. 2. **Rewrite the equation:** Let $y = e^{x}$. Then $e^{2x} = (e^{x})^2 = y^2$. Substit

1. **State the problem:** Solve the exponential equation $$x^{2} 2^{x} - 2^{x} 13 = 0.$$\n\n2. **Rewrite the equation:** Factor out the common term $$2^{x}$$ to get:\n$$2^{x}(x^{2}

1. **Problem:** Match each function with its corresponding graph. 2. **Recall the properties of exponential functions:**

1. **Problem 1:** Solve the equation $$\frac{x+5}{3} + \frac{x-3}{2} = \frac{x+7}{3}$$ 2. **Problem 2:** Solve the equation $$\frac{3}{x-5} + \frac{4}{x+7} = \frac{5}{x-6}$$

1. **State the problem:** Simplify the expression $$\frac{1}{4} \times 21^2 \times \frac{12}{11}^2$$. 2. **Rewrite the expression:** The expression can be written as $$\frac{1}{4}

1. **Problem Statement:** We have two functions describing water flow in a tank over time $t$ from 0 to 7.

1. **Problem statement:** A person travels at 5 km/hr until the remaining distance equals the time traveled in hours. Then, they increase speed to 8 km/hr and complete the journey

1. **Problem Statement:** Determine if the function $f(x) = x^4 - x^2$ has a horizontal asymptote. 2. **Recall the definition:** A horizontal asymptote is a horizontal line $y = L$

1. **State the problem:** Express the rational function $$\frac{x^3 - 2x}{x^2 + 3x + 2}$$ as partial fractions. 2. **Factor the denominator:** The denominator is a quadratic polyno

1. **Stating the problem:** We are given multiple algebraic expressions and asked to analyze or simplify them. These include quadratic polynomials, rational expressions, and quarti

1. The problem asks to find 80 percent of 450,000. 2. To find a percentage of a number, use the formula: $$\text{Percentage value} = \frac{\text{percent}}{100} \times \text{total v

1. The problem asks for the characteristic (integer part of the logarithm base 10) of the number 87950000. 2. The characteristic of a positive number $N$ is the integer part of $\l

1. **Problem Statement:** We are asked to analyze and factor or describe the given polynomial expressions:

1. **Problem Statement:** We are given three conditions for numbers: X is an even number, Y is a multiple of 3, and Z is less than 4. 2. **Understanding the conditions:**

1. **Problem Statement:** Find the difference between the total number of students in school P and school S.

1. **State the problem:** We want to find the sum of $x$ plus $x$. 2. **Use the formula:** When adding like terms, we add their coefficients. Here, both terms are $x$, which is the

1. **Stating the problem:** The present age of the father is the square of his son's age. We need to find after how many years the son's age will be half of the present age of the

1. **Problem Statement:** The present age of the father is the square of the present age of his son. We need to find after how many years the son's age will be half of the present