🧮 algebra

1. **Problem Statement:** We are given the 8th term ($T_8$) and the 17th term ($T_{17}$) of a Harmonic Progression (HP) as $\frac{2}{5}$ and $\frac{2}{11}$ respectively. We need to

1. **Problem Statement:** We need to find and indicate the solution region for the system of linear inequalities: $$3x + 7y \leq 21$$

1. Stating the problem: We need to find the simplified form of the quotient of two expressions: $$\frac{x^2 - 5x}{x + 2} \div \frac{3x - 15}{x^2 - 4}$$

1. The problem is to find the sum of the first $n$ terms of the series $1\cdot 2 + 2\cdot 3 + 3\cdot 4 + \cdots$. 2. Each term in the series can be written as $k(k+1)$ where $k$ is

1. **State the problem:** We are given the 11th term ($a_{11}$) and 19th term ($a_{19}$) of an arithmetic progression (AP) as 42 and 74 respectively. We need to find the sum of the

1. **Problem:** Determine if the function $f(x) = x^{-1} + x^7 + x^{-3}$ is even, odd, or neither. 2. **Recall definitions:**

1. The letter $t$ in math often represents a variable, commonly used to denote time in equations and functions. 2. It is a placeholder that can take on different values depending o

1. The problem states that the uppercase letter $R$ is 34 and asks to transfer it to the lowercase letter $r$ which is 17. 2. This suggests a relationship or conversion between $R$

1. The problem is to analyze the sign (positive, negative, or zero) of six different graphs or functions. 2. To determine the sign of a function, we look at the values of the funct

1. **Stating the problem:** Simplify the expression $x\sqrt{x}$. 2. **Recall the rule:** The square root of $x$ is $\sqrt{x} = x^{\frac{1}{2}}$.

1. **State the problem:** We have two pipes filling a swimming pool. The larger pipe runs for 4 hours, the smaller pipe for 9 hours, and together they fill half the pool. The small

1. **Determine whether each equation is linear in the given variables.** Linear equations in variables $x_1, x_2, x_3$ or $x, y$ have the form $$a_1 x_1 + a_2 x_2 + a_3 x_3 = b$$ o

1. **State the problem:** Solve the inequality $\frac{x+3}{2} \geq 5$. 2. **Formula and rules:** To solve inequalities involving fractions, multiply both sides by the denominator (

1. **State the problem:** Solve the inequality $$\frac{x}{9} - 6 > 4$$. 2. **Isolate the variable term:** Add 6 to both sides to get rid of the -6.

1. **Problem statement:** A number is first increased by 25% and then decreased by 20%. We need to find the net percentage change. 2. **Formula and explanation:**

1. **State the problem:** We need to find the percentage increase when a company's sales increase from 250 to 300. 2. **Formula for percentage increase:**

1. **Problem statement:** A can finish the entire task alone in 12 hours. We need to determine if A can finish half the task in 6 hours. 2. **Formula and concept:** The work rate o

1. **Problem statement:** Two persons start walking from the same point in opposite directions. One walks at 4 km/hr and the other at an unknown speed $x$ km/hr. We need to find th

1. The problem states that $3A = 2B = 6C$. We need to find the ratio $A : B : C$. 2. Let the common value be $k$, so:

1. **State the problem:** Solve the inequality $\frac{x}{2} + 1 \leq 5$. 2. **Formula and rules:** To solve inequalities, we isolate the variable on one side. Remember, when multip

1. **مقارنة العددين $2\sqrt{7}$ و $7\sqrt{2}$** نبدأ بحساب قيمة كل عدد تقريبا: