🧮 algebra

1. **State the problem:** We are given a piecewise function: $$g(x) = \begin{cases} 2 - x & \text{if } x \neq 2 \\ 1 & \text{if } x = 2 \end{cases}$$

1. **State the problem:** Solve the linear equation $6z + 9 = 28$ for $z$. 2. **Formula and rules:** To solve for $z$, isolate the variable by performing inverse operations. Subtra

1. **State the problem:** Determine the domain and range of the function $$f(x) = \frac{1}{x^2 - 4}$$. 2. **Domain:** The domain consists of all real numbers $x$ for which the func

1. **State the problem:** We are given two foci of a hyperbola at points $\left(2, 3 - \sqrt{25}\right)$ and $\left(2, 3 + \sqrt{25}\right)$, and the equations of its asymptotes: $

1. The problem is to solve the cryptarithm equation: $$ABB + BBA = CAB$$ where each letter represents a unique digit. 2. The goal is to find digits for $A$, $B$, and $C$ such that

1. **State the problem:** Simplify the expression $$\frac{x^2 - 12xy + 20y^2}{x^2 - 4y^2} \div \frac{3x - 6y}{x + 2y}$$. 2. **Rewrite division as multiplication:** Dividing by a fr

1. The problem is to simplify the expression $ad - bc$. 2. This expression is already in its simplest form as it is a difference of two products.

1. **State the problem:** Solve for $x$ in the equation $\log(x + 8) - \log 9 = \log 11$. 2. **Recall the logarithm subtraction rule:** $\log a - \log b = \log \left(\frac{a}{b}\ri

1. **State the problem:** There are 270 students in year 7 studying French, German, or Spanish. We know: - $\frac{2}{9}$ of the students study French.

1. **State the problem:** Expand the logarithmic expression $$\log \left( \frac{2(x+7)^5}{\sqrt[3]{x^4}} \right)$$ using properties of logarithms. 2. **Recall the properties of log

1. **State the problem:** Solve for $v$ in the equation $$1 = \sqrt{4v - 23} - 2$$ where $v$ is a real number. 2. **Isolate the square root term:** Add 2 to both sides to isolate t

1. **State the problem:** A homeowner wants to resod a front lawn that measures 10 \frac{1}{2} yards long and 6 \frac{2}{3} yards wide. The landscaper charges 8 per square yard for

1. **State the problem:** Solve the system of linear equations: $$-3x + 3y = 4$$

1. **State the problem:** Solve the quadratic equation by completing the square for $$x^2 - 10x + 19 = 0$$. 2. **Recall the method:** To complete the square, rewrite the quadratic

1. **State the problem:** Find the equation of the hyperbola with foci at $(-1, -1 - \sqrt{20})$ and $(-1, -1 + \sqrt{20})$ and asymptotes $y = \frac{1}{2}x - \frac{1}{2}$ and $y =

1. The problem gives two equations: $$KES + EES + SEE + MEES = 6543$$

1. **State the problem:** We want to find the maximum value of the function $$z = 4x + 3y$$ subject to the constraints:

1. **State the problem:** Solve the literal equation $$3n - 5p + 2n = 10p$$ for $$n$$. 2. **Combine like terms:** On the left side, combine $$3n$$ and $$2n$$ to get $$5n$$.

1. **State the problem:** Evaluate the expression $$\frac{5i}{3 - i}$$ where $i$ is the imaginary unit with the property $i^2 = -1$. 2. **Formula and rules:** To simplify a complex

1. **State the problem:** Simplify the expression $\frac{5i}{3 - i}$. 2. **Formula and rules:** To simplify a complex fraction with a complex denominator, multiply numerator and de

1. **State the problem:** Bill Pierson buys a 125-foot roll of latex tubing for 35 and sells all of it for 57.50. We need to find Bill's profit per foot, rounded to the nearest cen