🧮 algebra

1. **State the problem:** Find the partial fraction decomposition of the rational expression $$\frac{10}{5x^2 - 11x + 2}$$. 2. **Factor the denominator:** We need to factor the qua

1. The problem asks for the correct form of the partial fraction decomposition of the rational expression with denominator $(x - 8)(x^2 + 8)$. 2. When decomposing a rational functi

1. **State the problem:** We need to find the cost of a snack-size bag of chips, a bottle of water, and a gallon of milk given the total cost and relationships between their prices

1. **State the problem:** We have 3 gallons of milk, 6 bottles of water, and 5 snack-size bags of chips. The total cost is 25.50 before tax. 2. **Define variables:** Let $c$ be the

1. **State the problem:** Simplify the expression $$\frac{y^2}{3} - \frac{2}{y}$$. 2. **Identify the common denominator:** To combine the terms, find a common denominator. The deno

1. **State the problem:** Solve the equation $\frac{Y}{4} = -16$ for $Y$. 2. **Formula and rule:** To isolate $Y$, multiply both sides of the equation by 4 because dividing by 4 is

1. **State the problem:** Find the general term (nth term) of the sequence 3, 8, 13, 18, ... 2. **Identify the pattern:** The sequence increases by 5 each time (8 - 3 = 5, 13 - 8 =

1. **State the problem:** Solve for $w$ in the equation $$-\frac{5}{w+2} = -8 - \frac{7}{w-1}.$$\n\n2. **Rewrite the equation:** Move all terms to one side to isolate fractions:\n$

1. **State the problem:** Solve the equation $10\log(x-4) + 10\log(x-2) = \log 3^3$ using algebraic methods. 2. **Recall logarithm properties:**

1. **State the problem:** Solve for $x$ in the equation $$1 = \sqrt{2x - 6} - 1.$$ 2. **Isolate the square root:** Add 1 to both sides to get $$1 + 1 = \sqrt{2x - 6}$$ which simpli

1. Énoncé du problème : Déterminer quelle affirmation parmi les propositions données est vraie concernant les propriétés des logarithmes. 2. Rappel des propriétés importantes des l

1. **State the problem:** We are given the function $F(x) = 3x^4 - 2x^2 + 7$ and we want to analyze or work with it. 2. **Identify the type of function:** This is a polynomial func

1. Let's start by stating the problem: We want to understand how to find the domain of a function. 2. The domain of a function is the set of all possible input values ($x$) for whi

1. **State the problem:** Kimberly wants to find the number of bus rides where the total cost of paying per ride equals the total cost of buying a bus pass plus a reduced per-ride

1. **State the problem:** We need to find Jada's rate of change (slope) of her weekly profit as a function of the amount of paprika sold. 2. **Recall the formula for slope:** The s

1. **State the problem:** We need to find the slope of the line representing Riley's distance over time during the first 5 minutes of the race and write an equation for distance $y

1. **State the problem:** We need to evaluate three fractions: $$\frac{18^3}{0.395}, \quad \frac{0.28 \times 37.4}{77.8}, \quad \frac{63.7 \times \sqrt{3.93}}{0.425}$$

1. The problem asks to identify the graph representing the inequality $x \leq 1$. 2. The inequality $x \leq 1$ means all values of $x$ that are less than or equal to 1.

1. The problem involves understanding and comparing the inequalities: $x < 80$, $x \geq 80$, and $x > 80$. 2. Inequalities describe ranges of values for $x$:

1. **Stating the problem:** Solve the equation $$s = \frac{s(z-1)}{\frac{1}{3}(z+1)}$$ for $s$ or simplify it. 2. **Rewrite the denominator:** The denominator is $$\frac{1}{3}(z+1)

1. The problem asks to write the linear equation $y + 1 = -3(x - 1)$ in slope-intercept form and simplify. 2. The slope-intercept form of a linear equation is given by: