🧮 algebra

1. **State the problem:** We need to find the vertical asymptotes of the rational function $$f(x) = \frac{x - 2}{x^2 + 4x - 5}$$. 2. **Recall the rule for vertical asymptotes:** Ve

1. **State the problem:** We want to rewrite the expression $$\frac{5}{x+2}$$ using the least common denominator (LCD) of all terms in the equation $$\frac{2}{x} + \frac{3-x}{6} =

1. **State the problem:** Simplify the expression $$\frac{170+50x}{x} \div \frac{120+30x}{x}$$. 2. **Recall the division of fractions rule:** Dividing by a fraction is the same as

1. **State the problem:** We want to add the rational expressions $\frac{5}{2x}$ and $\frac{3x}{x + 1}$ which have different denominators.

1. **State the problem:** Simplify the rational expression $$\frac{x^2 - 9x + 20}{x^2 - 3x - 10}$$. 2. **Recall the formula and rules:** To simplify a rational expression, factor b

1. **State the problem:** Find the domain of the rational expression $$\frac{(x + 2)(x + 1)}{(x - 1)(x - 3)}$$. 2. **Recall the rule for domain of rational expressions:** The domai

1. **State the problem:** Find the inverse function $f^{-1}(x)$ of the function $f(x) = \sqrt{4x - 9}$. 2. **Recall the definition of inverse function:** The inverse function $f^{-

1. The problem asks us to identify which graphs represent a one-to-one function. 2. A function is one-to-one (injective) if and only if each output value corresponds to exactly one

1. **State the problem:** We need to find the value of the composite function $f(g(-1))$ given $f(x) = x^2 - 2x + 1$ and $g(x) = x + 2$. 2. **Recall the definition of composite fun

1. **State the problem:** We need to find the value of $f(3)g(3)$ given the functions $f(x) = 3x^2 - 5$ and $g(x) = -2x + 5$. 2. **Evaluate $f(3)$:** Substitute $x=3$ into $f(x)$:

1. **State the problem:** We need to find the value of $f(-1) + g(-1)$ given the functions $f(x) = 2x^2 + 3x - 1$ and $g(x) = -x$. 2. **Recall the formula:** To find $f(a)$ for som

1. **State the problem:** We need to find the value of the function $f(x) = 2(x^2 + 5)$ when $x = -3$. 2. **Recall the formula:** The function is given by $$f(x) = 2(x^2 + 5).$$

1. **State the problem:** We are given the quadratic equation $$y = 2x^2 - 8x + 3$$ and asked to describe its graph. 2. **Identify the type of graph:** This is a quadratic function

1. **State the problem:** We want to divide the polynomial $$x^3 - x^2 + 0x + 12$$ by the binomial $$x - 3$$ using synthetic division. 2. **Identify the divisor root:** For synthet

1. **State the problem:** We want to find the area of a rectangle with height $2x + 5$ and width $12 + 3x$ in terms of the variable $x$. 2. **Formula for area of a rectangle:**

1. **State the problem:** We need to find which quadratic equation among the four given has no real solution. 2. **Recall the discriminant formula:** For a quadratic equation $ax^2

1. **State the problem:** Simplify the expression $$\frac{1 - 2i}{3 - 2i}$$ as a single complex number. 2. **Formula and rule:** To divide complex numbers, multiply numerator and d

1. **State the problem:** We need to solve the quadratic equation $$x^2 + 3x - 10 = 0$$ by factoring. 2. **Recall the factoring method:** To factor a quadratic equation of the form

1. **State the problem:** We need to identify the graph that represents the quadratic inequality $$y \geq x^2 - 6x + 11$$. 2. **Rewrite the quadratic expression:** The quadratic ca

1. **State the problem:** We need to solve the quadratic inequality $$x^2 - 14x + 48 \leq 0$$. 2. **Find the roots of the quadratic equation:** First, solve the equation $$x^2 - 14

1. **State the problem:** We want to find the time intervals when the height $y$ of the projectile is less than 20 feet. 2. **Write the inequality:**