🧮 algebra

1. **State the problem:** Coach Leo scheduled \(\frac{2}{5}\) hours for dribbling drills and \(\frac{1}{4}\) hour for team scrimmage. The gym is available for 1 hour. We need to de

1. Stating the problem: Solve the inequality $$\frac{1}{3^x} + 1 + \frac{1}{3^x} - 3 \leq 0$$. 2. Combine like terms: $$\frac{1}{3^x} + \frac{1}{3^x} = 2 \cdot \frac{1}{3^x}$$.

1. **Stating the problem:** Solve the inequality $$\left(\frac{1}{3^x}+1\right) + \frac{1}{3^x} - 3 \leq 0.$$\n\n2. **Rewrite the inequality:** Combine like terms involving $\frac{

1. The problem is to simplify the expression $\left(\frac{1}{3^x}+1\right)+\frac{1}{3^x}-3$. 2. Rewrite the expression as $\frac{1}{3^x} + 1 + \frac{1}{3^x} - 3$.

1. Stating the problem: Solve the equation $$2^x - 2^{-x} = \frac{15}{4}$$. 2. Let us set $$y = 2^x$$. Then, $$2^{-x} = \frac{1}{2^x} = \frac{1}{y}$$.

1. The problem asks us to find the modulus (absolute value) of the result of \(8 \div 4\).\n 2. First, perform the division inside the modulus: \(8 \div 4 = 2\).\n

1. The problem states ages of Jack (10 years), Kylie (17 years), and Vanessa (23 years), and a total amount of £16 shared between Kylie and Vanessa in the ratio of their ages. 2. F

1. We are given two inequalities: $$x^2 + y^2 \leq 4$$ and $$ (x + 2)^2 + y^2 \leq 4.$$

1. State the problem: Find the cost to print 100,000 copies given the linear cost function $C = Ax + B$ where the cost for 10,000 copies is 5000 and for 15,000 copies is 6000. 2. U

1. Stated problem: Solve the inequality \( \frac{3}{X-1} < \frac{2}{X+1} \). 2. To solve the inequality, first eliminate the fractions by multiplying both sides by the common denom

1. **Stating the problem:** We need to find all pairs of positive integers $(n,m)$ such that the sum of factorials from $1!$ to $n!$ equals a perfect square $m^2$, i.e., $$1! + 2!

1. **State the problem:** We need to find a real root of the cubic function $$f(x) = 9.34 - 2.19x + 16.3x^2 - 3.7x^3.$$ This means solving for $x$ such that $f(x) = 0$. 2. **Rewrit

1. We are given integers $m$ and $n$ with $1 < m \leq n$ and function $$ f(m,n) = \left(1 - \frac{1}{m}\right) \left(1 - \frac{1}{m+1}\right) \cdots \left(1 - \frac{1}{n}\right). $

1. **State the problem:** We are given three positive real numbers $a,b,c$ such that $$\frac{a}{a+b} = \frac{a+b}{a+b+c} = \frac{c}{b+c}$$

1. The problem is to understand the equation of a plane or line in three variables: $3x - 8y - 7z = 58$. 2. This is a linear equation in the variables $x$, $y$, and $z$. It represe

1. We are given two quadratic equations: $$2x^2 - mx + 8 = 0$$ with roots $$\alpha$$ and $$\beta$$.

1. প্রথমে সমীকরণটি লিখো: $$\frac{1}{x} = \frac{1}{4a} + \frac{1}{4b}$$ 2. সমীকরণের ডান পাশে ভগ্নাংশগুলোর লিসি মূল করো: $$\frac{1}{x} = \frac{b}{4ab} + \frac{a}{4ab} = \frac{a+b}{4a

1. **Problem:** For each pair of functions, identify one characteristic they share and one characteristic that distinguishes them. 2. **Part a) Functions:** $f(x)=\frac{1}{x}$ and

1. We are asked to classify each given function as one of the following types: power function, root function, polynomial (stating its degree), rational function, algebraic function

1. We are asked to evaluate each expression without using a calculator. 2. (a) Evaluate $(-3)^4$:

1. Stating the problem: We have three positive real numbers $a$, $b$, $c$ such that $$\frac{a}{a+b} = \frac{a+b}{a+b+c} = \frac{c}{b+c}$$