🧮 algebra

1. **State the problem:** Marcy burns 80% of the calories walking compared to jogging. We want to write a function for calories burned walking as a function of calories burned jogg

1. The problem asks whether the left-hand side (LHS) of an equation or expression is positive or negative at $x=0$. 2. To determine this, we need the explicit form of the LHS funct

1. The problem asks to write the left-hand side (LHS) of an inequality evaluated at a specific point $x_0$. 2. Generally, an inequality can be written as $f(x) \leq g(x)$ or $f(x)

1. **State the problem:** We want to solve the inequality $$(x-3)(x^2+1)(x^2+2x-15) > 0.$$ 2. **Find the zeros (roots) of the polynomial:** The zeros occur where each factor equals

1. **State the problem:** A car rental agency charges 20 per day to rent a car. We need to write the cost as a function of the number of days rented and find the cost for 5 days. 2

1. **State the problem:** Calculate how much a waitress makes for a 4-hour and an 8-hour shift given the pay function $P(t) = 2.52t + 78$, where $t$ is the number of hours worked.

1. **Problem:** Simplify the expression $12 + 14\frac{1}{2} + \frac{1}{4}21 + 41$. 2. **Step 1:** Convert mixed numbers and fractions to improper fractions or decimals for easier a

1. **State the problem:** We need to evaluate the expression $$Tr = \frac{5^2}{\pi^2 \times \alpha} \ln \left[ \frac{8}{\pi^2} \times \frac{240 - 60}{105 - 60} \right]$$ where $\al

1. **State the problem:** We need to simplify and evaluate the expression $$T_r = \frac{5^2}{\pi^2 \times \alpha} \ln \left[ \frac{8}{\pi^2} \times \frac{240-60}{105-60} \right]$$

1. **Problem statement:** Mary's grandparents invest an amount $x$ on 1st January 1998 and continue to deposit $x$ on the first day of each month. The account pays 0.4% interest pe

1. **State the problem:** Solve the equation $$\log_3 \sqrt{x} = \frac{1}{2 \log_2 3} + \log_3 (4x^3)$$ where $$x > 0$$. 2. **Recall logarithm properties:**

1. **Problem Statement:** We need to clarify the expression and solve or simplify it based on the correction: the term is $4x^3$ at the end, not $2$, and there is no $x$ after the

1. **State the problem:** Solve the equation $$\log_3 \sqrt{x} = \frac{1}{2 \log_3 x} + \log_3 (4x^2)$$ where $$x > 0$$. 2. **Recall logarithm properties:**

1. **State the problem:** Given the value of $n$ as 14, we want to understand or use this value in a mathematical context. 2. **Explanation:** Since the problem only states "n is 1

1. **State the problem:** We have two numbers, 27 and 36, and we want to show that their least common multiple (LCM) is 108 by grouping blocks in tables of 108 blocks. 2. **Recall

1. **State the problem:** We are given the midpoint $M(-3,-1)$ of segment $\overline{AB}$ and one endpoint $A(-2,5)$. We need to find the coordinates of the other endpoint $B(x,y)$

1. **State the problem:** We are given the midpoint $M(-4,4)$ of the line segment $\overline{AB}$ and one endpoint $A(-6,1)$. We need to find the coordinates of the other endpoint

1. **State the problem:** Simplify the expression $$\frac{5ef \cdot y}{10ef - 5fg}$$. 2. **Identify the formula and rules:** We will factor the denominator and simplify the fractio

1. مسئله را بیان می‌کنیم: تابع $f(x) = x + \sqrt{x^3 + 1}$ داده شده است و وارون آن به صورت $f^{-1}(x) = ax + \frac{b}{x}$ برای $x > 0$ است. هدف یافتن حاصل $ab$ است. 2. برای یافتن و

1. **State the problem:** We are given the midpoint $M(-2,-3)$ of the line segment $\overline{AB}$ and one endpoint $A(-8,-5)$. We need to find the coordinates of the other endpoin

1. مسئله را بیان می‌کنیم: باید مقدار $$g^{-1}(f^{-1}(8))$$ را پیدا کنیم که در آن $$f(x) = \frac{4x + 3}{2x - 4}$$ و $$g(x) = \frac{x^2 + 1}{4}$$. 2. ابتدا معکوس تابع $$f$$ را پیدا