🧮 algebra

1. **Problem Statement:** We want to find pairs of numbers from the set $\{1, 2, 2, 2, 3, 4, 4, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 8, 8, 9\}$ that can produce the numbers in the set $\{

1. The problem is to understand how to factor algebraic expressions. 2. Factoring means rewriting an expression as a product of simpler expressions.

1. The problem asks for the domain of the function $(q \circ r)(x)$, which means the composition of functions $q$ and $r$. 2. The domain of $(q \circ r)(x)$ is all $x$ values in th

1. **State the problem:** We are given two functions: $$q(x) = \frac{1}{x - 8}$$

1. The problem is to add the fractions $\frac{1}{5}$ and $\frac{2}{5}$.\n\n2. When adding fractions with the same denominator, you add the numerators and keep the denominator the s

1. **State the problem:** Simplify the expression $$\frac{2}{3} \times \frac{4}{5}$$. 2. **Recall the rule for multiplying fractions:** Multiply the numerators together and the den

1. The problem asks to predict the amount after 50 days using the best fitting curve, which is given as Figure 3: $$y = 546(0.98)^x$$. 2. The formula for exponential decay is $$y =

1. **State the problem:** Solve the inequality $$(x + 2)(x - 3)(x + 3)(x + \geq 0$$ (assuming the last factor is $(x + 1)$ due to typo). 2. **Rewrite the inequality:**

1. **State the problem:** Given $a = \frac{1+\sqrt{5}}{2}$, find the value of $a^6 - 8a^2$. 2. **Recall the formula and properties:** The number $a = \frac{1+\sqrt{5}}{2}$ is the g

1. **State the problem:** Find the domain of the function $$y = -\sqrt{x}$$. 2. **Recall the domain rule for square root functions:** The expression inside the square root must be

1. **State the problem:** Find the domain of the function $$y = \sqrt[3]{x - 1}$$. 2. **Recall the domain rule for cube root functions:** The cube root function $$\sqrt[3]{x}$$ is

1. **State the problem:** Solve the equation $$\frac{1}{x + 6} + \frac{x}{x - 6} = \frac{2}{x^2 - 36}$$ for $x$. 2. **Recall the formula and rules:** The denominator $x^2 - 36$ can

1. **State the problem:** Solve the rational equation $$\frac{7}{x + 2} + \frac{3x + 5}{x - 2} = \frac{2x + 7}{x - 2}$$. 2. **Identify the denominators:** The denominators are $x+2

1. **Problem Statement:** Multiply the fractions given in Part 4. 2. **Formula:** To multiply fractions, multiply the numerators together and the denominators together:

1. The problem is to find three different equations that have the same solution as the equation $2x + 9 = -15$. 2. First, solve the original equation to find the solution for $x$.

1. **State the problem:** Simplify the expression $$\frac{x^4 + 7x^3}{2x^2 + 14x}$$. 2. **Identify common factors:**

1. **State the problem:** We need to solve the equation $2x + 3 = 7$ for $x$. 2. **Formula and rules:** To solve a linear equation, isolate the variable on one side by performing i

1. **Problem:** Find the value of $x + y$ when $x=2$ and $y=1$. 2. **Formula:** Use simple addition: $x + y$.

1. **State the problem:** Simplify the expression $$\frac{5x^2}{20x^4 - 120x^3}$$. 2. **Factor the denominator:** Factor out the greatest common factor (GCF) from the denominator.

1. **State the problem:** Add the two fractions $$\frac{3}{b-8} + \frac{7}{b+3}$$. 2. **Formula and rules:** To add fractions, find a common denominator, which is the product of th

1. **State the problem:** We have a sequence of patterns where each pattern number $n$ contains $n$ hexagons and $n$ triangles. 2. **Identify the pattern:** For pattern number 1, t