🧮 algebra

1. **State the problem:** Solve the equation $$\frac{2x+3}{4} = 5$$ for $x$. 2. **Formula and rules:** To solve for $x$ in a fraction equation, multiply both sides by the denominat

1. **Problem statement:** Find the compositions $[f \circ g](x)$ and $[g \circ f](x)$ for the functions: $f(x) = x + 5$ and $g(x) = x - 3$

1. **State the problem:** Solve the inequality $2 - 3a < -8$ for $a$. 2. **Isolate the term with $a$:** Subtract 2 from both sides:

1. **State the problem:** Solve the inequality $$-13 \geq 2(b - 6) + 3$$ for $b$. 2. **Apply the distributive property:**

1. Das Problem: Du möchtest verstehen, wie Quotientengleichheit und Produktgleichheit funktionieren. 2. Quotientengleichheit bedeutet, dass wenn zwei Brüche gleich sind, also $$\fr

1. **State the problem:** We need to find the cost per issue of a monthly magazine subscription that costs 15.96 for one year. 2. **Identify the formula:** Since the subscription i

1. **State the problem:** We need to factor the polynomial $$w^6 - 7w^3 + 10$$. 2. **Rewrite the polynomial:** Notice that $$w^6 = (w^3)^2$$, so let $$x = w^3$$. The polynomial bec

1. The problem asks to find the expression equivalent to $36x^2 - 169$. 2. Recognize that $36x^2 - 169$ is a difference of squares because $36x^2 = (6x)^2$ and $169 = 13^2$.

1. The problem asks to find which expression is equivalent to $50x^2 - 128$. 2. We start by factoring the expression $50x^2 - 128$.

1. The problem asks us to find an expression equivalent to $x^3 - 125$. 2. Recognize that $x^3 - 125$ is a difference of cubes since $125 = 5^3$.

1. The problem states the relationship $d = \sqrt{\frac{3h}{2}}$ where $d$ is a distance and $h$ is a height with $h \geq 0$. 2. This formula models how distance $d$ depends on hei

1. **State the problem:** We start with the parent function $f(x) = \sqrt[3]{x}$. We want to find the new function $g(x)$ after three transformations: - Reflection over the x-axis

1. **Problem:** Determine if the proportion $\frac{4}{5} = \frac{20}{25}$ is true. 2. **Formula:** Two fractions $\frac{a}{b}$ and $\frac{c}{d}$ are equal if $ad = bc$.

1. **State the problem:** Simplify the expression $$(10x^0)^6$$. 2. **Recall the rules:**

1. **State the problem:** Simplify the expression $x^2 \cdot x \cdot x^6$. 2. **Recall the rule for multiplying powers with the same base:** When multiplying powers with the same b

1. **State the problem:** We are given the polynomial function $$f(x) = -5x^3 - 30x^2 + 200x$$ and want to analyze it.

1. **State the problem:** Simplify or analyze the quadratic expression $-4x^2 + 16x + 84$. 2. **Identify the formula and rules:** This is a quadratic expression of the form $ax^2 +

1. The problem involves simplifying expressions with square roots and variables: $$4.5\sqrt{3524} - 3xyx^4 - 4hy$$ and $$5\sqrt{8xy} + xhy^3$$. 2. Recall the properties of square r

1. **State the problem:** Given that $a - b = b - c = 2$, find the value of $$\frac{(a - b)^2 + (b - c)^2}{(a - c)^2}$$

1. The problem is to make the left side look like the original state of the right side. 2. To solve this, we need to identify the expressions on both sides and apply algebraic oper

1. The problem is to solve an equation or expression where the right side must remain unchanged. 2. When solving equations, the goal is to isolate the variable on one side without