🧮 algebra

1. The problem is to express a number or expression as a fraction. 2. A fraction is a number expressed as $\frac{a}{b}$ where $a$ and $b$ are integers and $b \neq 0$.

1. **State the problem:** Solve the equation $5 - x = 14$ for $x$. 2. **Formula and rules:** To solve for $x$, isolate $x$ on one side of the equation by performing inverse operati

1. **State the problem:** Simplify the expression $$(10c^6 d^{-5})(2c^{-5} d^4)$$ and find which option it matches. 2. **Write the formula and rules:** When multiplying expressions

1. The problem is to simplify and verify the expression $\left(9^{1/2}\right)^2$ and identify the correct simplification among the options. 2. Recall the exponent rule: $\left(a^m\

1. **State the problem:** Hassan and Raju initially have the same number of points. Hassan exchanges 624 points for a prize. After this, Raju has 5 times as many points as Hassan.

1. **Problem Statement:** Factor completely the expressions:

1. The problem asks to identify the system of inequalities that represents a rectangular shaded region centered on the origin, symmetric about the x-axis, extending horizontally al

1. **State the problem:** We are given the function $$h(x) = - \frac{2}{3} x - 5$$ and need to find the value of its inverse function at 1, i.e., $$h^{-1}(1)$$. 2. **Recall the def

1. **State the problem:** We need to determine which graph corresponds to the system of inequalities: $$y \geq 3x + 2$$

1. **State the problem:** Find the inverse of the function $$y = \sqrt[4]{x} - 2$$. 2. **Recall the definition of inverse functions:** The inverse function swaps the roles of $$x$$

1. **State the problem:** We need to find the region in the xy-plane that satisfies the system of inequalities: $$y \geq \frac{3}{2}x + 2$$

1. The problem is to identify the correct inverse function $f^{-1}(x)$ from the given options. 2. To find the inverse function, recall that if $y = f(x)$, then $x = f^{-1}(y)$, mea

1. **State the problem:** We are given the function $$y = \sqrt[3]{x} + 2$$ and want to understand its graph and behavior. 2. **Formula and rules:** The cube root function is defin

1. The problem asks to identify which system of inequalities could represent the graph shown in the xy-plane. 2. The inequalities involve either $y$ or $x$ with boundary lines at $

1. **State the problem:** Find the inverse of the function $$f(x) = x^2 - 8$$ with the domain restriction $$x \geq 0$$. 2. **Recall the definition of inverse function:** The invers

1. **State the problem:** We are given the quadratic function $$p(t) = -1.875 t^2 + 30 t - 20$$ which represents profit as a function of time $$t$$. 2. **Identify the type of funct

1. **State the problem:** Find the inverse of the function $f(x) = 3x - 6$. 2. **Recall the formula for inverse functions:** To find the inverse $f^{-1}(x)$, swap $x$ and $y$ in th

1. **State the problem:** We are given the equation $$2x^2 - 2x = 112$$ and need to find which of the provided options is equivalent to it. 2. **Rewrite the equation in standard fo

1. **State the problem:** We are given the equation $$-3x^2 - 9x = -84$$ and need to find an equivalent equation from the options provided. 2. **Rewrite the equation:** To find an

1. **State the problem:** Determine if the functions $y=\frac{1}{3}x+3$ and $y=-3x-3$ are inverses of each other. 2. **Recall the rule for inverse functions:** Two functions $f$ an

1. The problem asks to find the inverse of the function $f(x)$ given by the table: | $x$ | $f(x)$ |