🧮 algebra

1. **Stating the problem:** We need to find the value of $x$ in an addition problem where the total was mistakenly taken as 540 instead of 431. 2. **Understanding the problem:** If

1. The problem involves evaluating the function $g(x)$ at specific points and understanding the range of $g$. 2. Part A states the range is $(-\infty, 4]$. This means $g(x)$ can ta

1. **State the problem:** Aurora baked some cookies. She gave 12 cookies to her family, sold \(\frac{7}{9}\) of the cookies at a bake sale, and had 8 cookies left over. We need to

1. The problem is to solve the inequality $$6x \geq 108$$ and find all possible values of $x$ that satisfy it. 2. The formula used here is to isolate $x$ by dividing both sides of

1. **State the problem:** We need to determine which values among I. 7.5, II. \frac{22}{3}, and III. 8 satisfy the inequality $$\frac{x}{3} > 2\frac{1}{2}$$. 2. **Rewrite the inequ

1. **State the problem:** Simplify and expand the expression $(x-2)^2$. 2. **Formula used:** The square of a binomial $(a-b)^2$ is given by $$ (a-b)^2 = a^2 - 2ab + b^2 $$

1. **State the problem:** Factor out the greatest common factor (GCF) from the polynomial $$5x^4 - 25x^3 + 10x^2$$. 2. **Identify the GCF:** Look at the coefficients 5, 25, and 10.

1. The problem is to factor the trinomial $x^2 - 12x - 45$ or determine if it is prime. 2. The general form of a quadratic trinomial is $ax^2 + bx + c$. Here, $a=1$, $b=-12$, and $

1. **State the problem:** Solve the system of equations by elimination: $$10x + 2y = 18$$

1. **Problema:** Resolver a inequação $$4^{-1} + 2x \leq \frac{1}{2} - 3x$$. 2. **Fórmula e regras:** Para resolver inequações, isolamos a variável $x$ em um lado da desigualdade,

1. **State the problem:** Simplify the complex fraction $$\frac{\frac{2}{x} + 7}{8}$$. 2. **Rewrite the numerator:** The numerator is $$\frac{2}{x} + 7$$. To combine these terms, w

1. **Problem statement:** We want to find the function term of a cubic polynomial $f(x)$ that models the curve profile of a swing lounger. The function is a cubic polynomial symmet

1. **Planteamiento del problema:** Simplificar la expresión $$\frac{x^2 - 2x - 3}{x^2 + x - 6} \div \frac{x - 3}{4x - 8}$$. 2. **Regla para dividir fracciones:** Dividir por una fr

1. **State the problem:** Evaluate the cube root of -81, i.e., find $\sqrt[3]{-81}$.

1. **State the problem:** Simplify the expression $$\frac{1 - \frac{4}{x+8}}{x + \frac{16}{x+8}}$$. 2. **Rewrite the expression:** To simplify, write the numerator and denominator

1. Problem: Solve the exponential equation $10^x = 100^{2x - 3}$ by finding a common base. 2. Recall that $100 = 10^2$, so rewrite the right side:

1. **State the problem:** We need to find the domain and range of the function $f$ based on the graph description. 2. **Recall definitions:**

1. **Problem:** Determine who has finished 4/5 of their project. 2. **Step 1: Calculate total fraction finished by each person by adding their parts.**

1. **State the problem:** Simplify the expression $$\frac{2 \cdot x \cdot x \cdot x - 8 \cdot x \cdot x \cdot x}{2 \cdot x}$$. 2. **Rewrite the expression:** The numerator is $$2x^

1. **State the problem:** We want to find the equation representing the combinations of 6-passenger cars ($x$) and 3-passenger cars ($y$) such that the total passengers per run is

1. The problem involves two inequalities: $$y \geq 3x + 1$$ and $$y \leq 3x - 3$$. 2. We want to find the solution set that satisfies both inequalities simultaneously.