🧮 algebra

1. **Problem statement:** Provide two alternative definitions for the set of rational numbers using the idea of ordered pairs. 2. **Definition 1:** Rational numbers can be defined

1. The problem is to analyze and graph the function $$y=\left(\frac{1}{4}\right)^{-x} - 5$$. 2. Recall the rule for negative exponents: $$a^{-x} = \frac{1}{a^x}$$. Applying this to

1. **State the problem:** We want to analyze the function $$y=|x|+\frac{1}{|\sin(4x)|}$$ which involves absolute values and a trigonometric function. 2. **Understand the components

1. Problem: Find the zeros of the function $$f(x) = \frac{(\ln x)^2}{2 - x}$$. 2. To find zeros of a function, set the numerator equal to zero and ensure the denominator is not zer

1. **Problem a:** Simplify $$\left(\frac{a^{2}}{3} \times \left[\frac{a^{3}}{1} \times \frac{1}{a^{4}}\right]\right)^{6 \times \frac{1}{7}}$$. 2. First, simplify inside the bracket

1. **State the problem:** We want to analyze the function $$y = |x| + \frac{1}{|\sin(4x)|}$$ and understand its behavior. 2. **Recall important rules:**

1. **State the problem:** Find the equation of a parabola whose axis is parallel to the x-axis and passes through points (3, 1), (0, 0), and (8, -4). 2. **Recall the form of the pa

1. **State the problem:** Simplify the expression $4x - (3x+2)(4x-1)$. 2. **Use the distributive property:** Expand the product $(3x+2)(4x-1)$ using the FOIL method:

1. The problem involves finding the product of the binomials $(4x + 1)(x + 1)$ and simplifying the expression. 2. The formula for multiplying two binomials is given by the distribu

1. The problem is to understand and analyze the function $y = |x| + \frac{1}{|skn|}$. 2. Here, $|x|$ denotes the absolute value of $x$, which means it is always non-negative regard

1. The problem asks which method is less messy when multiplying two polynomials each with 3 terms: length $x^2 + 2x + 5$ and width $5x^2 + 3x - 7$. 2. The two common methods to mul

1. The problem is to analyze the function $$g(x) = |x| + |\sin(4\pi)|.$$\n\n2. First, recall that the absolute value function $|x|$ returns the non-negative value of $x$.\n\n3. Nex

1. **State the problem:** Solve the linear equation $5 - 2y = 7$ for $y$. 2. **Use the formula and rules:** To solve for $y$, isolate $y$ on one side by performing inverse operatio

1. **Problem statement:** Find the graph of the function $$g(x) = |x| + \frac{1}{\sin(4\pi)}$$.

1. **State the problem:** We are given the height of a rocket as a function of time: $$h = -2(t+1)(t-9)$$ and need to answer several questions about the rocket's height and timing.

1. **State the problem:** Solve the equation $$-6 = \frac{p}{4} - 6$$ for $p$. 2. **Add 6 to both sides** to isolate the term with $p$:

1. **State the problem:** You sold 300 shirts at 20 each. For every 1 increase in price, sales drop by 5 shirts. Find the number of shirts to sell to maximize revenue and the maxim

1. **Énoncé du problème** : Trouver le point d'intersection entre les fonctions $y_1 = 4\sqrt{x}$ et $y_2 = 2x + 2$, puis déterminer les bornes d'intégration entre ces courbes et l

1. The problem is to identify the vertical asymptote (VA) and the possible presence of a horizontal or oblique (slant) asymptote (CA) for a function with denominator $x+1$. 2. A ve

1. **State the problem:** We need to find the equation of a rational function with a point of discontinuity (hole) at $(-3,-5)$, a $y$-intercept at $(0,4)$, and an $x$-intercept at

1. **State the problem:** We need to sketch the graph of the piecewise function: