🧮 algebra

1. **State the problem:** We have a curve given by $$y = x^2 + 2cx + 4$$ and a line given by $$y = 4x + c$$, where $$c$$ is a constant. We want to find the values of $$c$$ for whic

1. **Problem statement:** (i) Show that two distinct lines $y = ax + c$ and $y = a'x + c'$ have a common point unless $a = a'$. Also show this when one line has infinite slope.

1. **Planteamiento del problema:** Calcular el dominio de la función $$f(x) = e^{\frac{3x}{x^2 - 1}} + \frac{2x}{x^2 + 9}$$.

1. **Problem:** Solve for $x$ in the equation $x - 5 = -8$ and then evaluate $(x^2 - 5)(x^3 - x)$. 2. **Step 1: Solve for $x$.**

1. **State the problem:** Given the equation $$R_G T_G + R_B T_B = 120$$ with $$R_G = 4$$, $$R_B = 10$$, and the relation $$T_G = T_B + 2$$, find the values of $$T_G$$ and $$T_B$$.

1. **State the problem:** Determine the domain of the function represented by a V-shaped graph centered at the origin (0,0). 2. **Understand the graph:** The graph is a V-shape wit

1. **State the problem:** We need to find the domain of the exponential function shown in the graph. 2. **Recall the domain of exponential functions:** The domain of any exponentia

1. The problem states that a 14-inch pepperoni pizza is sliced into 8 equal pieces, and one slice contains 352 calories. 2. We want to find the total calories $c$ in the whole pizz

1. **State the problem:** Calculate the value of $1.005^{96}$. 2. **Formula used:** The expression is an exponentiation where the base is $1.005$ and the exponent is $96$.

1. مسئله: تبدیل عدد اعشاری 1.083 به کسر عام مارت. 2. ابتدا عدد را به صورت جمع عدد صحیح و قسمت اعشاری می‌نویسیم: $$1.083 = 1 + 0.083$$

1. **Problem statement:** We have a line with equation $y = mx + c$ and a curve with equation $xy = 16$.

1. **Problem 9:** For each function, find the greatest domain, range, and whether it is one-to-one or many-to-one. **a) Function:** $f(x) = 4x - 5$

1. **Problem:** For the function $f(x) = 4x - 5$, find: i) The greatest possible domain

1. **State the problem:** Solve the equation $x + 1 = \sqrt{7x + 15}$. 2. **Understand the equation:** The left side is linear, and the right side is a square root function. To sol

1. **State the problem:** Solve the equation $$(2x)^{\frac{3}{4}} + 2 = 10$$ for $x$. 2. **Isolate the term with the exponent:** Subtract 2 from both sides:

1. **State the problem:** Solve the equation $$\sqrt[3]{x} - 9 = 2$$ for $x$. 2. **Isolate the cube root:** Add 9 to both sides to isolate the cube root term.

1. **State the problem:** We are given a system of inequalities: $$5x + y \leq 10$$

1. **State the problem:** We need to graph the inequality $3x + 4y \geq 12$ on the Cartesian plane. 2. **Rewrite the inequality as an equation to find the boundary line:**

1. **State the problem:** We are given a one-to-one function $$f(x) = \frac{5x - 7}{7x + 5}$$ and need to find its inverse function $$f^{-1}(x)$$.

1. **State the problem:** Solve the quadratic equation $$6(-4m - 5)^2 + 1(-4m - 5) - 1 = 0$$ for $m$. 2. **Substitute:** Let $x = -4m - 5$ to simplify the equation.

1. **State the problem:** Solve the quadratic equation $$(3x + 3)^2 + 6(3x + 3) + 5 = 0.$$\n\n2. **Use substitution:** Let $y = 3x + 3$. The equation becomes $$y^2 + 6y + 5 = 0.$$\