🧮 algebra

1. **State the problem:** We are given a polynomial function $$f$$ defined on the domain $$-4 \leq x \leq 4$$, which is an odd function. The table provides values and behavior of $

1. **State the problem:** We want to find the intervals where the rational function $$r(x) = \frac{x^3 + 4x^2 + 4x}{x^2 - 9}$$ is greater than or equal to zero, i.e., $$r(x) \geq 0

1. **State the problem:** We are given the complex number $6+3i$. 2. **Find the complex conjugate:** The complex conjugate of a complex number $a+bi$ is $a - bi$.

1. The problem asks us to determine if the given equation defines $y$ as a function of $x$. 2. A relation defines $y$ as a function of $x$ if for every value of $x$, there is exact

1. **State the problem:** We need to determine if the given equation defines $y$ as a function of $x$. 2. **Recall the definition of a function:** A relation defines $y$ as a funct

1. The problem is to calculate the factorial of 3, denoted as $3!$. 2. The factorial of a positive integer $n$ is the product of all positive integers from 1 to $n$. The formula is

1. **State the problem:** A hall requires $1125$ floor tiles. Tiles are sold only in full boxes, each containing $48$ tiles. Each box costs $18.75$ and delivery costs $24.60$. We n

1. **State the problem:** We need to graph the equation $$-2y = x - 4$$ and identify the x- and y-intercepts. 2. **Rewrite the equation in slope-intercept form:** Solve for $$y$$.

1. **State the problem:** Find the equation of the line passing through the points $(-2, 5)$ and $(-1, 10)$ in slope-intercept form and then write it in function notation. 2. **Fin

1. **State the problem:** We need to graph the piecewise function $$g(x) = \begin{cases} 1 & \text{if } x < 0 \\ -2 & \text{if } x = 0 \\ -1 & \text{if } x > 0 \end{cases}$$ 2. **U

1. The user asked to solve problems 30, 31, and 32, but only problems 23 and 24 are provided.\n2. Since problems 30, 31, and 32 are not given, I cannot solve them.\n3. According to

1. The problem is to isolate the variable $x$ in the equation $$x + 9 = -12$$ using only one step. 2. To isolate $x$, we need to undo the addition of $9$. The inverse operation of

1. **State the problem:** Solve the equation $$6\left(y - \frac{1}{3}\right) = 4(3y - 5)$$ for $$y$$. 2. **Distribute both sides:**

1. **State the problem:** We need to find the volume of a rectangular tank where:

1. The problem is to evaluate $\ln\left(\frac{2}{1}\right)$.\n\n2. Recall the logarithm property: $\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)$.\n\n3. Applying this property, we

1. **State the problem:** Solve for $x$ in the equation $$x^2 = -2(3 - 2)$$. 2. **Simplify the right side:** Calculate inside the parentheses first:

1. **State the problem:** Solve the equation $$\sqrt{x-6} - x = -6$$ for $$x$$. 2. **Isolate the square root:** Add $$x$$ to both sides to get:

1. **State the problem:** We need to determine which of the given equations represents a parabola that opens down. 2. **Recall the rule for parabolas:**

1. **State the problem:** We need to find the correct equation of a parabola given its graph. 2. **Identify key features:** The parabola opens downward and has its vertex at the po

1. The problem is to solve the equation involving a square root and identify any extraneous solutions. 2. Since the equation is incomplete as given ($\sqrt{\ } + 12 =$), we need th

1. The problem is to understand and simplify the expression $-\frac{i\sqrt{15}}{3}$. 2. This expression is a complex number in the form $a + bi$, where $a=0$ and $b=-\frac{\sqrt{15