🧮 algebra

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

1. **State the problem:** We are given the function $$f(x) = \frac{x+1}{6x^2 - 7x - 3}$$ and need to determine whether the points $$x=\frac{3}{2}, x=-\frac{1}{3}, x=-\frac{3}{2},$$

1. State the problem: Simplify fully the expression $3(2x - 5) - 4(x + 1)$.\n\n2. Apply the distributive property: Multiply each term inside the parentheses by the factor outside.\

1. Problem statement: The price of an item is 3000 in the year 2023. The price is expected to increase by 3% per year. We define $x$ as the number of years after 2023 and $p(x)$ as

1. **State the problem:** Solve the compound inequality $$-4 \leq \frac{1}{2}(8x + 8) < 12$$ and find the solution set. 2. **Recall the rule:** When solving inequalities, you can m

1. **State the problem:** We have a Fibonacci-type sequence where each term is the sum of the two previous terms. The sequence is: $a_1$, 5, $a_3$, $a_4$, 23, ...

1. **State the problem:** Simplify fully the expression $3(2x - 5) - 4(x + 1)$. 2. **Apply the distributive property:** Multiply each term inside the parentheses by the factor outs

1. **Stating the problem:** We need to find the value of the sum $$\frac{1}{\sqrt{1} + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \cdots + \frac{1}{\sqrt{35} + \sqrt{36}}$$

1. **State the problem:** Simplify the expression $$\frac{(3m^{-2} n)^{-3}}{6mn^{-2}}$$ assuming $$m \neq 0$$ and $$n \neq 0$$. 2. **Recall the rules:**

1. **State the problem:** Solve the inequality $$2 - \frac{m}{11} \leq -7$$ and determine the solution set. 2. **Isolate the variable term:** Subtract 2 from both sides:

1. **State the problem:** We need to find the percentage decrease in the world record time between February 2013 and March 2013. 2. **Convert times to seconds:**

1. **State the problem:** We need to find the starting value of Hattie's house given that after one year it increased by 9%, and after the second year it decreased by 4%, resulting