🧮 algebra

1. **State the problem:** Solve the equation $$\frac{1}{2}(x+2) - \frac{1}{4}(x-4) = \frac{1}{8}(x+4)$$ for $x$. 2. **Distribute the fractions:**

1. The problem is to understand what the natural logarithm function \(\ln\) means and why \(\ln(1)=0\), \(\ln(e)=1\), and \(\ln(e^2)=2\). 2. The natural logarithm \(\ln(x)\) is the

1. The problem is to simplify the expression by factoring out the common multiplier: $$500 \times 79 - 250 \times 157$$. 2. Identify the common multiplier between 500 and 250. The

1. **Aufgabe 2: Potenzen von 2 und -5** Gegeben sind Potenzen von 2 und -5 mit verschiedenen Exponenten. Wir berechnen jede Potenz einzeln.

1. **State the problem:** Find the Highest Common Factor (HCF) and Least Common Multiple (LCM) of the polynomials $16a^4 - 4a^2 - 4a - 1$ and $8a^3 - 1$. 2. **Factorize the first p

1. **State the problem:** Prove that 6 is a factor of $n(n^2 + 5)$ for all integers $n \geq 1$ using mathematical induction. 2. **Base case:** For $n=1$, evaluate $1(1^2 + 5) = 1 \

1. **State the problem:** We need to prove that 6 is a factor of the expression $n(n^2 + 5)$ for all integers $n \geq 1$. 2. **Rewrite the expression:** The expression is $n(n^2 +

1. **State the problem:** A man's salary in 1975 was 2000 Naira per year, and it increased by 10% each year. We need to find the total amount he earned from 1975 to 1984 inclusive.

1. The problem is to find the square root of 13. 2. The square root of a number $x$ is a value $y$ such that $y^2 = x$.

1. The problem is to add two irrational numbers and simplify the result. 2. First example: $\sqrt{3} + (-\sqrt{3}) = \sqrt{3} - \sqrt{3} = 0$.

1. **State the problem:** We are given the function $$y = \log_2 |x - 2|$$ and asked to analyze it. 2. **Understand the function:** This is a logarithmic function with base 2, wher

1. The problem asks to determine if each table of values represents a function. 2. A relation is a function if every input $x$ has exactly one output $y$.

1. Problem: Determine if the mapping from domain \{0,1,2,5,6\} to range \{5,6,7,8\} with arrows 0→5, 1→7, 2→6, 5→7, 6→8 is a function. Step: Each domain element maps to exactly one

1. **State the problem:** Determine whether each set of ordered pairs, graph points, tables, and mapping diagrams represent functions. ---

1. The first diagram maps each domain element to exactly one range element: 0→5, 1→6, 2→7, 5→8, 6 has no mapping shown. Since no domain element maps to more than one range element,

1. **State the problem:** Determine if each set of ordered pairs, graph points, tables, and mapping diagrams represent functions. ---

1. The problem is to find the remainder when -41 is divided by 9, i.e., compute $-41 \bmod 9$. 2. Recall that the modulo operation finds the remainder after division, and the resul

1. **Stating the problems:** We have four equations involving the variable $P$:

1. **State the problem:** We have two functions: $$f(x) = -\frac{1}{2}x^2 + x + \frac{3}{2}$$

1. The problem asks to express the given numbers in standard form, which means writing them as a product of a number between 1 and 10 and a power of 10. 2. For part (a), 51720:

1. **State the problem:** We are given the graph of a function $f$ and need to estimate: