🧮 algebra

1. The problem is to understand and analyze the formula for $N$ given by $$N = \frac{6000}{5p + z}.$$\n\n2. This formula expresses $N$ as a function of two variables $p$ and $z$. T

1. **Stating the problem:** We want to find the valid formula for the value $y$ at time $t$ during the interval $0 \leq t \leq T$, given that $y$ starts at $y_s$ and decreases to $

1. **Problem statement:** A company reduces the price of a product by 20% and sales income increases by 20%. Costs per product remain the same, no fixed costs, and total profit rem

1. **Problem statement:** Simplify the expression $$\frac{(x+9)^2}{x} \cdot \frac{x^2}{2x + 18}$$ 2. **Formula and rules:** When multiplying fractions, multiply numerators together

1. **State the problem:** We need to find the number of male adults who are not active members in a sports club in a district. 2. **Given data:**

1. **Problem Statement:** Determine the domain and range of the absolute value function graphed with vertex at $(-2,0)$, decreasing from $(-5,3)$ to $(-2,0)$, and increasing from $

1. **State the problem:** We are given three ratios involving four sizes $X_1, X_2, X_3,$ and $X_4$: $$X_1 : X_2 = 2 : 3$$

1. **State the problem:** Determine the domain and range of the absolute value function graphed, and find the intervals where the graph is increasing and decreasing.

1. The problem states that the share price of a company falls by 60%. We want to find the percentage increase needed to return to the original price. 2. Let the original price be $

1. **State the problem:** Simplify the expressions involving square roots: $\sqrt{143}$, $7\sqrt{105}$, $6\sqrt{105}$, and $210\sqrt{3}$.\n\n2. **Recall the simplification rule:**

1. **State the problem:** We are given the transformation of the absolute value function $f(x) = |x|$ to $g(x) = |x + 3| - 5$ and asked to solve the equation $|6x + 3| = 33$. We al

1. **State the problem:** Patrick, Marcel, Anthony, and Yannick caught 35 fishes in total. Marcel caught three times as many fishes as Yannick. Patrick caught three more fishes tha

1. **Problem statement:** Berechne und kürze das Ergebnis der Multiplikation der Brüche so weit wie möglich. 2. **Formel:** Bei der Multiplikation von Brüchen multipliziert man Zäh

1. The problem is to solve the equation $|2x + 5| = 19$ for $x$. 2. Recall that the absolute value equation $|A| = B$ implies two cases: $A = B$ or $A = -B$, provided $B \geq 0$.

1. **State the problem:** Lavonne gave tickets and snack vouchers to her work group members. Each member received the same number of tickets and the same number of snack vouchers.

1. The problem asks us to find the formula for the function $g(x)$ which is a transformation of $f(x) = |x|$ shifted left 3 units and down 5 units. 2. The general rule for horizont

1. **State the problem:** We have a machine that fills a form of dimensions 50 cm wide, 1 m high, and 1.5 m long in 15 minutes. We want to find how long 10 machines take to fill a

1. We are asked to solve the quadratic equation $$x^2 + 2x - 6 = 0$$ for $x$. 2. The quadratic formula to find roots of $$ax^2 + bx + c = 0$$ is:

1. **State the problem:** We want to express the rational function $$\frac{x^4+3x^3+4x^2+3x+1}{x(x^2+1)^2}$$ as a sum of partial fractions.

1. **State the problem:** Mr. Yong bought caps and T-shirts. The ratio of caps to T-shirts is 3:2.

1. **State the problem:** We have a rectangle with original length $L$ and width $W$. When the length is increased by 2 cm, the area becomes 36 cm². When the width is increased by