🧮 algebra

1. **Problem statement:** Given the graph of $f$ with x-intercept $(5,0)$ and y-intercept $(0,8)$, find: (a) The y-intercept of $f(x) + 3$.

1. **State the problem:** Simplify the expression $$\frac{2x^2}{x^2} - \frac{x^2 + 25x}{x^2}$$. 2. **Recall the rule:** When subtracting fractions with the same denominator, subtra

1. The problem asks to find Keenan's bank account balance when he buys 9 raffle tickets based on the given graph. 2. From the graph description, the balance decreases linearly from

1. **State the problem:** Simplify the expression $$\frac{x^2 + x}{x + 10} + \frac{10x + 10}{x + 10}$$. 2. **Identify the common denominator:** Both fractions have the same denomin

1. The problem asks: If the airplane flew 35 miles, how many gallons of gas would be left in the tank? 2. From the graph description, the gas remaining decreases linearly from abou

1. **Problem:** Factorize the quadratic equation $$z^2 + (4 - 2i)z + (5 - 4i) = 0$$ by completing the square. 2. **Formula and rules:** To complete the square for a quadratic equat

1. **Problem:** Given $f(x) = 3x - 3$ and $g(x) = \frac{2}{x-1}$, find $\left(\frac{f}{g} - 2f(3)\right)$. 2. **Step 1:** Calculate $f(3)$.

1. **State the problem:** We are given two numbers:

1. **Problem statement:** It takes 12 hours to fill a swimming pool using 7 identical taps. We want to find how many hours it would take to fill the same size pool using only 4 of

1. **State the problem:** Simplify the expression $$\frac{x^2 + 3x}{x + 10} - \frac{3x + 100}{x + 10}$$. 2. **Identify the formula and rules:** Since both fractions have the same d

1. **State the problem:** A teacher gave each student 4 purple cards and 3 white cards. The total number of cards given out is 91. We need to find:

1. **State the problem:** We know the elevator travels 330 feet in 10 seconds. We want to find how far it travels in 11 seconds. 2. **Identify the formula:** Since the elevator mov

1. **Problem Statement:** Prove by mathematical induction that for all positive integers $n$:

1. We are asked to solve the inequality $$2 < \frac{2x + 3}{x + 1} < \frac{5}{2}$$ for real numbers $x$ such that $x > 1$. 2. Important: Since $x > 1$, the denominator $x + 1 > 0$,

1. The problem is to find the value of the function $y = 2x^2 - 3x + 5$ when $x = 4$. 2. The formula given is a quadratic function: $$y = 2x^2 - 3x + 5$$ where $x$ is the input var

1. **State the problem:** Simplify the expression $$\frac{x^2 - 36}{x + 4} \cdot \frac{4x}{6x - 36}$$. 2. **Recall important formulas and rules:**

1. **Problem statement:** Simplify the expression $2x + 2 - 3x - 3 + 6x^2 - 6$. 2. **Formula and rules:** Combine like terms by adding or subtracting coefficients of the same power

1. **State the problem:** Evaluate the expression $$\sqrt[3]{3k} + 5k$$ when $$k = -72$$. 2. **Recall the formula and rules:** The cube root of a number $$a$$ is a value $$b$$ such

1. The problem is to understand and analyze the function $y = x^2$. 2. This is a quadratic function, which generally has the form $y = ax^2 + bx + c$. Here, $a=1$, $b=0$, and $c=0$

1. **State the problem:** Simplify the expression $$\frac{5x}{2y} \div \frac{3x}{4y}$$. 2. **Recall the rule for division of fractions:** Dividing by a fraction is the same as mult

1. The problem is to understand and express the general form of the $n$th root of $x$, written as $\sqrt[n]{x}$. 2. The $n$th root of a number $x$ is the number that, when raised t