🧮 algebra

1. **State the problem:** We want to analyze and understand the function $$f(x) = \sqrt{2x^2 - 9x - 5}$$ and determine where it is defined and its graph shape. 2. **Domain of the f

1. The problem appears to be a sequence of numbers: 2,400,000; 2,100,000; 1,800,000; 1,500,000; 1,200,000; 900,000. 2. We want to identify the pattern or rule governing this sequen

1. Planteamos el problema: Encontrar la ecuación de la recta que pasa por el punto $\left(\frac{1}{2}, -\frac{2}{3}\right)$ y es perpendicular a la recta dada $9x - 5y = 1$. 2. Pri

1. **State the problem:** We are given a quadratic profit function $P = ax^2 + bx + c$ where $x$ is the number of items produced and $P$ is the weekly profit. We have three data po

1. Stating the problem: Factorize the expression $$E = x^2 - 2x - 15$$. 2. Use the quadratic expression factorization formula: For $$ax^2 + bx + c$$, find two numbers that multiply

1. Problema: Hallar la ecuación de la recta que pasa por el punto $\left(\frac{1}{2}, -\frac{2}{3}\right)$ y es perpendicular a la recta dada por $9x - 5y = 1$. 2. Primero, encontr

1. **Stating the problem:** We need to redo calculations so that the total sum equals 9,800,000. 2. **Understanding the problem:** Suppose the original calculation was a sum of sev

1. **Problem statement:** Given the equation $$\frac{ب = م (2 - س ) - (7 - 4)}{3} = 1 = \frac{م}{6 - س - س^2} 2$$ and asked to find the value of $م$. 2. **Rewrite the equation cle

1. نبدأ بكتابة المعادلة المعطاة: $$d(s) = -s^2 + 4s + m$$ حيث نريد إيجاد قيمة $m$ ومساحة المربع $ص ا م$. 2. نفهم أن المربع موضوع بحيث أحد رؤوسه على محور $s$ (أي $d(s) = 0$ عند هذا

1. **State the problem:** We are given that six tablets of soap cost 440. We need to find the cost of 11 tablets. 2. **Identify the type of proportion:** The problem involves direc

1. **State the problem:** Simplify the expression $(-12 + 7) - (-5)$. 2. **Apply the order of operations:** First, simplify inside the parentheses.

1. **Problem statement:** Find the percent error for Sam's foot measurement. 2. **Formula:** Percent error is calculated as

1. **Stating the problem:** We are given the system:

1. **Stating the problem:** Solve the inequality involving absolute values: $$|x-1| < |x+1|$$ and analyze the solution intervals. 2. **Recall the definition of absolute value:** Fo

1. **State the problem:** Solve the equation $3y = 7 - \frac{2}{5}$ for $y$. 2. **Rewrite the equation:** The equation is $3y = 7 - \frac{2}{5}$.

1. **State the problem:** Solve for $x$ in the equation $$\sqrt{\frac{(x+2)!}{x!}} = \sqrt{3!} \times 7.$$\n\n2. **Recall factorial and square root properties:** The factorial $n!$

1. **State the problem:** We need to find how many solutions the equation $d + 2 + 5d = 2 + 6d$ has. 2. **Write the equation:**

1. **State the problem:** Solve the equation $-8w = -6(2w - 8)$ and determine the number of solutions. 2. **Write the equation:**

1. The problem is to find the value of $x$ using proportions given as $\frac{40}{800} = \frac{30}{x}$. 2. The formula for solving proportions is to cross-multiply: $a/b = c/d$ impl

1. **State the problem:** We need to find the cost to deliver a container a distance of 30 km given the cost-distance relationship shown by the graph. 2. **Identify known points:**

1. **State the problem:** We are given an arithmetic sequence with first term $a_1 = 2$ and common difference $d = 7$. We need to find the 12th term of this sequence. 2. **Formula