🧮 algebra

1. **State the problem:** Rationalize the denominator of the fraction $$\frac{1}{\sqrt[3]{5} + \sqrt[3]{7} + 2}$$ where the denominator has two cube roots and one rational part. 2.

1. The problem is to graph the piecewise function: $$y=\begin{cases}2x & 0 \leq x \leq 4 \\ 10 & 4 < x \leq 8\end{cases}$$

1. The problem is to simplify the fraction $\frac{4}{9}$ and express it as a decimal. 2. To convert a fraction to a decimal, divide the numerator by the denominator.

1. **State the problem:** We want to understand why the linear system $$\begin{cases} 3x + y = 10 \\ 2x - 2y = 8 \end{cases}$$

1. Let's understand why we multiply by $-2$ in certain problems. 2. Often, multiplying by $-2$ is used to eliminate a variable or to simplify an equation.

1. **State the problem:** We need to solve the system of linear equations graphically: $$y = x + 1$$

1. **State the problem:** We need to find the ratio of gold to silver to bronze medals won by Switzerland in the 2010 Winter Olympic Games. 2. **Identify the values:** From the tab

1. **State the problem:** Solve the inequality $$\frac{8}{3} u < -12$$ for $u$. 2. **Formula and rules:** To solve inequalities involving multiplication or division by a negative n

1. The problem is to solve the inequality $$-5 > \frac{y}{3}$$ for $y$. 2. To isolate $y$, we multiply both sides of the inequality by 3. Since 3 is positive, the inequality direct

1. **State the problem:** We are given two functions $f(x) = 2x + 4$ and $g(x) = 3x^2$. We need to find the function operation $(f + g)(x)$ and then determine the domain of the res

1. **State the problem:** We are given two functions $f(x) = 2x + 4$ and $g(x) = 3x^2$. We need to find the function operation $(f+g)(x)$ and then determine the domain of the resul

1. **State the problem:** We are given the function $f(x,y) = x^2 y + \cos(x)$ and need to understand or analyze it. 2. **Identify the components:** The function has two variables

1. **State the problem:** Solve the quadratic equation $$x^2 + 18x + 62 = 0$$ by completing the square. 2. **Recall the formula and rule:** To complete the square for an equation o

1. **State the problem:** Solve the quadratic equation $$x^2 - 14x + 38 = 0$$ by completing the square. 2. **Recall the formula and rule:** To complete the square for an equation o

1. **State the problem:** There are 90 kids in the band. 20% of the kids own their own instruments, and the rest rent them. We need to find how many kids own their own instruments.

1. **State the problem:** We need to find how many milligrams of vitamin C are in 1 orange, given that 1 orange has about 75% of the recommended daily allowance (RDA) of 45 mg. 2.

1. **State the problem:** Prove by induction that for all natural numbers $n$, the inequality $$\frac{1}{5^{n+1}} + \frac{1}{5^{n+2}} + \cdots + \frac{1}{2 \cdot 5^n} > \frac{1}{2}

1. **State the problem:** Prove by induction that for all natural numbers $n$, the inequality $$\frac{1}{5^{n+1}} + \frac{1}{5^{n+2}} + \cdots + \frac{1}{2 \cdot 5^n} > \frac{1}{2}

1. **State the problem:** We want to find the transformation that converts the graph of $f(x) = -x^2 + 9$ into the graph of $g(x) = -4x^2 + 9$. 2. **Compare the functions:** Both f

1. **State the problem:** We need to solve the inequality $$(x^2 - x - a) \cdot \left(\sqrt{x^2 - 4x + 4} - a\right) > 0$$ for each value of the parameter $a$. 2. **Rewrite and ana

1. **State the problem:** We need to solve the inequality $$(x^2 - x - a) \cdot \left(\sqrt{x^2 - 4x + 4} - a\right) > 0.$$