🧮 algebra

1. **State the problem:** Vani wants to treat 12 friends with either bubble tea or ice-cream cones. 2. **Define variables:** Let $x$ be the number of bubble teas and $y$ be the num

1. **State the problem:** Solve the compound inequality $$d-5 < \frac{2d}{5} \leq \frac{d}{2} + \frac{1}{5}$$. 2. **Break the compound inequality into two parts:**

1. **State the problem:** We have two types of energy drinks: 2-Liter jugs costing 4 each and 0.5-Liter bottles costing 1.5 each.

1. **State the problem:** We have two types of energy drinks: 2-Liter jugs costing 4 each and 0.5-Liter bottles costing 1.5 each.

1. **State the problem:** Solve the simultaneous equations: a) $y = 4x - 10$

1. **State the problem:** Solve the system of linear equations: $$0.5x + 2y = 10$$

1. **Problem statement:** Given the family of functions defined for $k > 0$ by $$f_k(x) = \frac{1}{2k} \cdot x^2 \cdot (x - 2k)^2,$$

1. **Problemstellung:** Löse das Gleichungssystem $$2^y = 2 \cdot 4^x$$

1. **Problem statement:** We have two functions \(y = x - |x - 1| + 2\) and \(y = |x|\) to plot on the same graph. Also, solve the inequality \(|x| + |x - 2| \leq x + 4\) and find

1. The problem states that we must use the answer from part one to solve the current problem. 2. To proceed, identify the answer from part one, denote it as $x$.

1. **State the problem:** We need to find the number of solutions to the equation $$x^4 = 3 - 20x^2$$.

1. **Problem statement:** Berechnen Sie die Nullstellen der quadratischen Funktion $h(x) = x^2 + 6x + 9$. 2. **Formel:** Die Nullstellen einer quadratischen Funktion $ax^2 + bx + c

1. The problem is to evaluate the logarithm of a fraction, specifically $\log \frac{2}{ }$. However, the denominator is missing, so the expression is incomplete. 2. The logarithm o

1. Problem: Löse die Gleichung $m + 5 = 11$ nach $m$ auf. 2. Formel: Um eine Variable zu isolieren, subtrahiere die Zahl auf der gleichen Seite wie die Variable von beiden Seiten d

1. **State the problem:** Solve the equation $$\frac{2x+4}{x-3} = 3$$ for $x$.

1. **Problem statement:** Alice wants to buy the same number of apples, oranges, and pears. Apples come in packs of 12, oranges in packs of 7, and pears in packs of 9. We need to f

1. **State the problem:** Solve the equation $$\frac{2x+4}{x-3} = 3$$ for $x$. 2. **Formula and rules:** To solve a rational equation, multiply both sides by the denominator to eli

1. **State the problem:** Solve the equation $2n + 6 = 14$ for $n$. 2. **Write down the equation:**

1. We are asked to express $x^2 + 4x + 1$ in the form $(x + a)^2 - b$ and then solve $x^2 + 4x + 1 = 0$. 2. To express $x^2 + 4x + 1$ in the form $(x + a)^2 - b$, we complete the s

1. **State the problem:** Simplify and verify the expression $$(\sqrt{3} + \sqrt{2})^2 - (\sqrt{3} - \sqrt{2})^2 = 4 \sqrt{6}$$. 2. **Recall the formula:** Use the difference of sq

1. **State the problem:** Simplify the expression $$(17^6)^5 \cdot (17^7)^{-8}$$ giving the answer with a positive exponent. 2. **Recall the exponent rules:**