🧮 algebra

1. **State the problem:** We are given the function $f(x) = x^{-2}$ and need to understand its behavior. 2. **Recall the formula and rules:** The function $f(x) = x^{-2}$ can be re

1. The problem states: Given $X=5$ and $Y=6$, find $(XY)^2$. 2. The formula to use is the power of a product: $$(XY)^2 = X^2 Y^2$$ which means you square each factor and then multi

1. **State the problem:** Solve the equation $5X - 10 = -15$ for $X$. 2. **Add 10 to both sides to isolate the term with $X$:**

1. **Problem Statement:** Jacob spends 60 minutes in the gym doing freehand exercises and running on the treadmill. He runs on the treadmill for 30 minutes longer than he does free

1. **State the problem:** We have two functions:

1. The problem asks for the closest distance a person can be to the number 6000. 2. To find the closest distance between two numbers, we use the absolute value of their difference:

1. **State the problem:** We need to find which values among 0, 7, and 6 satisfy the inequality $$-1 - 2x \leq -10$$. 2. **Solve the inequality:**

1. **State the problem:** Given the height function $$h = -5t^2 + 290t + 600$$, find the height at $$t=10$$, the times when height is zero, and rewrite the function in vertex form.

1. **State the problem:** Solve the inequality $-4 + 2x \geq 1$. 2. **Add 4 to both sides:**

1. **Statement of the problem:** We have a sequence $(u_n)$ defined by $u_0 = -1$ and the recurrence relation $$u_{n+1} = \frac{3}{2} u_n - 1.$$ We define another sequence $(v_n)$

1. **Problem Statement:** Simplify the expression $\frac{7x^2}{4} - 3$. 2. **Formula and Rules:** This is a simple algebraic expression involving a fraction and subtraction. No spe

1. **State the problem:** We are given an arithmetic progression (A.P.) where the sum of the 5th and 12th terms is 93, and the 8th term is 43. We need to find the first four terms

1. Simplify each expression and express with positive exponents. **a)** Simplify $\left(\frac{a^3}{b^{-4}}\right)^{-2}$

1. **Énoncé du problème :** Simplifier l'expression $$\ln\left(\frac{1}{2}\right) + \ln\left(\frac{2}{3}\right) + \ln(5)$$. 2. **Formule utilisée :** La somme des logarithmes de pr

1. **Simplify:** $\ln\left(\frac{1}{2}\right) + \ln\left(\frac{2}{3}\right) + \ln(5)$ 2. Use the logarithm property: $\ln(a) + \ln(b) = \ln(ab)$.

1. حل المعادلة $2x + 3 = 7$: $$2x + 3 = 7$$

1. نُعطى الشعاعين $$\mathbf{u} = \begin{pmatrix}1 \\ 3\end{pmatrix}$$ و $$\mathbf{v} = \begin{pmatrix}z^2 - 1 \\ z + 1\end{pmatrix}$$ ونريد إيجاد قيمة العدد الحقيقي $$w$$ بحيث $$\m

1. **State the problem:** Expand and fully simplify the expression $y(6y + 2) + 5(2y + 4)$. 2. **Use the distributive property:** Multiply each term inside the parentheses by the f

1. **State the problem:** Expand and simplify the expression $2(w - 1) + 5(2w + 3)$. 2. **Use the distributive property:** Multiply each term inside the parentheses by the factor o

1. **State the problem:** Simplify the expression $10x^2 - (x-2)x^2 - (x+8)$. 2. **Distribute and expand:** Apply the distributive property to the terms inside the parentheses:

1. Problem: Write equations for the given relationships. - w is directly proportional to v.