🧮 algebra

1. **State the problem:** Solve the inequality $$\frac{x - 1}{3} > 3x - \frac{4}{3}$$. 2. **Write the inequality clearly:**

1. **Problem statement:** Factor the quadratic expressions given in part 21, starting with (a) 4y^2 - 20y - 56. 2. **Formula and rules:** To factor a quadratic expression of the fo

1. **Problem:** Simplify the expression $$(5x - 2)(x + 4) - (2x + 1)(5x - 3) + 5x^2.$$ 2. **Formula and rules:** Use distributive property (FOIL) to expand each product, then combi

1. The problem is to rationalize the expression, which means to eliminate any radicals from the denominator of a fraction. 2. The general formula for rationalizing a denominator wi

1. Problem: For the function $f(x) = -2x^3$, analyze the behavior as $x \to -\infty$. The leading term dominates the behavior of the polynomial for large $|x|$. Since the leading t

1. **State the problem:** Simplify the expression $$\frac{12}{2 - \sqrt{3}}$$. 2. **Formula and rule:** To simplify expressions with radicals in the denominator, multiply numerator

1. **State the problem:** Multiply the polynomials $ (x + 6)(x - 2) $ and simplify the result. 2. **Formula used:** To multiply two binomials, use the distributive property (also k

1. **State the problem:** Multiply the polynomials $ (v - 5)(v + 4) $ and simplify the result. 2. **Recall the formula:** To multiply two binomials, use the distributive property (

1. **State the problem:** Simplify each expression involving squares and square roots. 2. **Recall the rule:** For any positive number $a$, $(\sqrt{a})^2 = a$ because squaring and

1. **State the problem:** Simplify the fraction $\frac{602}{20}$. 2. **Formula and rules:** To simplify a fraction, divide the numerator and denominator by their greatest common di

1. **State the problem:** We analyze the rational function $$f(x) = \frac{3x^2 - 13x - 10}{x^2 - 2x - 15}$$ to find points of discontinuity, vertical and horizontal asymptotes, and

1. **State the problem:** Solve for $x$ in the equation $$\frac{6}{x+2} = \frac{1}{4} + \frac{x-7}{x+2}.$$

1. **State the problem:** We are given the area of a playground as 504 square yards. The length is expressed as $x + 8$ and the width as $x - 2$. We need to find the values of leng

1. **State the problem:** We need to find the polynomial that represents the area of a rectangle whose length is 5 inches longer than its width. 2. **Define variables:** Let the wi

1. The problem is to understand the decimal 0.90 and its equivalent fraction form. 2. We know that decimals can be converted to fractions by considering the place value. Here, 0.90

1. The problem is to convert the fraction $\frac{9}{10}$ into different equivalent forms and understand its decimal and percentage representations. 2. The fraction $\frac{9}{10}$ m

1. **State the problem:** Simplify the fraction $\frac{667}{28}$ if possible. 2. **Check for common factors:** To simplify a fraction, find the greatest common divisor (GCD) of the

1. **State the problem:** Multiply the mixed numbers $5 \frac{3}{4}$ and $4 \frac{1}{7}$. 2. **Convert mixed numbers to improper fractions:**

1. **State the problem:** Solve for $x$ in the equation $$\left(\frac{4 \frac{1}{5}}{x} + \frac{1}{3}\right) \div 2 \frac{4}{35} - \frac{4}{5} = 1 \frac{8}{15}.$$ 2. **Convert mixe

1. **State the problem:** Simplify the expression $$0.5 - [7 - 2 - (1 - 9) - 3 + 12] + 4.$$\n\n2. **Recall the order of operations:** Parentheses first, then brackets, then additio

1. The problem asks to explain why there are two possible values for $a$ in the equation $x^2 + y^2 = 100$ and to find these values. 2. The equation $x^2 + y^2 = 100$ represents a