🧮 algebra

1. **Problem Statement:** We are given two functions:

1. **State the problem:** Solve the equation $\frac{b}{5} = \frac{8}{16}$ for $b$. 2. **Use the cross-multiplication method:** When two fractions are equal, their cross products ar

1. The problem is to convert a quadratic equation from vertex form to standard form. 2. The vertex form of a quadratic equation is given by $$y = a(x-h)^2 + k$$ where $(h,k)$ is th

1. **Stating the problem:** We want to find the equation of a parabola given its graph. 2. **General form of a parabola:** The standard form of a parabola that opens up or down is

1. **State the problem:** We want to find how many gallons of yellow paint are needed to make 48 gallons of green paint, given that the mixture ratio is 5 gallons of blue paint to

1. **Problem A:** Determine the truth of the biconditional "if $\sqrt{x} \geq x$ if and only if $0 < x < 1$." - The functions involved are $y = \sqrt{x}$ and $y = x$.

1. The user asks: "Okay but that points do i graph?" which implies a question about which points to plot for a graph. 2. To answer this, I need to know the function or equation to

1. **State the problem:** We need to understand and graph the piecewise function: $$f(x) = \begin{cases} -3x - 12 & \text{for } x \leq -3 \\ 1 & \text{for } x > 5 \end{cases}$$

1. **State the problem:** Simplify the expression $$\left(\frac{-12x^{-5}y}{4x^{3}}\right)^{-2}$$. 2. **Recall the rules:**

1. **State the problem:** Simplify the expression $$\frac{-12x^{-5}y}{(4x^3)^{-2}}$$. 2. **Recall the rule for negative exponents:** For any nonzero number $a$ and integer $n$, $$a

1. **State the problem:** We want to express the function $$R(x) = x - 8$$ as a composition of three functions $$f \circ g \circ h$$, where none of the functions are the identity f

1. **State the problem:** Verify the order of the rational numbers $\frac{4}{11}$, $\frac{6}{17}$, $\frac{2}{5}$, $\frac{3}{7}$, $\frac{5}{12}$, and $\frac{3}{8}$ from least to gre

1. **State the problem:** Order the rational numbers $\frac{4}{11}$, $\frac{6}{17}$, $\frac{2}{5}$, $\frac{3}{7}$, $\frac{5}{12}$, and $\frac{3}{8}$ from least to greatest. 2. **Me

1. **State the problem:** We are given the equation $x + 12 = 6y$ and need to complete the table of values for $x$ and $y$. 2. **Rewrite the equation to solve for $y$:**

1. **State the problem:** Complete the table for the equation $x + 12 = 6y$ and verify the points. 2. **Rewrite the equation in slope-intercept form:**

1. The problem asks to evaluate $\left(\frac{4}{5}\right)^{-2}$.\n\n2. Recall the rule for negative exponents: $a^{-n} = \frac{1}{a^n}$. This means we take the reciprocal of the ba

1. **State the problem:** Evaluate the expression $$\left( \frac{r^3 s^{-1}}{r^2 s^6} \right)^{-1}$$ for $$r=8$$ and $$s=-2$$. 2. **Simplify the expression inside the parentheses:*

1. **State the problem:** Solve the equation $$5 + 6(x + 9) = 59 - 9(3x + 1)$$ and check the solution. 2. **Expand both sides:**

1. **State the problem:** Find the cube root of the expression $$87448p^9r^2$$. 2. **Recall the formula:** The cube root of a product is the product of the cube roots: $$\sqrt[3]{a

1. **State the problem:** Subtract the polynomials $\left(5x^2 - 9x + 4\right)$ and $\left(-x^2 + 6x + 2\right)$. That is, find $$\left(5x^2 - 9x + 4\right) - \left(-x^2 + 6x + 2\r

1. **State the problem:** Simplify the expression $$9t^2 - 7t + 2 + (3t + 10)$$ and verify if it equals $$2x^2 - 9x - 20$$. 2. **Combine like terms:** Add the polynomials by combin