🧮 algebra

1. **State the problem:** Simplify the expression $$\frac{x-2}{x+2} + \frac{x+10}{x^2+6x+8}$$. 2. **Factor the denominator:** Note that $$x^2+6x+8$$ factors as $$(x+2)(x+4)$$.

1. **State the problem:** Find the point of intersection (POI) of the two lines given by the equations: $$6x + 2y = -6$$

1. **Problem:** Solve the equation $\log x = 3$. 2. **Recall the definition:** $\log x$ means logarithm base 10, so $\log x = 3$ means $10^3 = x$.

1. The problem asks to find the product of $\sqrt{b} \cdot \sqrt{b}$ assuming $b \geq 0$. 2. Recall the property of square roots: $\sqrt{x} \cdot \sqrt{y} = \sqrt{xy}$ for $x,y \ge

1. Das Problem lautet: "Mach einfacher". 2. Um einen Ausdruck zu vereinfachen, suchen wir nach gemeinsamen Faktoren, kürzen Brüche oder fassen Terme zusammen.

1. **State the problem:** Multiply the expressions $$\sqrt{5x^8y^2} \times \sqrt{10x^3} \times \sqrt{12y}$$ assuming $$x \geq 0$$ and $$y \geq 0$$. 2. **Recall the property of squa

1. **State the problem:** Simplify the expression $$\frac{15x^2}{x^2 + 7x - 18} - \frac{6x^5}{x^2 - 11x + 18}$$ and analyze its components. 2. **Factor the denominators:**

1. **State the problem:** Simplify the expression $$\frac{x - 2}{x + 2} + \frac{x + 10}{x^2 + 6x + 8}$$. 2. **Factor the denominator:** Notice that $$x^2 + 6x + 8$$ can be factored

1. **State the problem:** Simplify the expression $$\frac{5x^9}{8x^{11} + x^2} - \frac{15x^2}{8x^7 + x - 18}$$. 2. **Analyze each term:** The expression is a subtraction of two rat

1. The problem is to factor the quadratic expression $r^2 - 12r + 35$ into the form $(r - a)(r - b)$. 2. We use the factoring method for quadratics: find two numbers $a$ and $b$ su

1. **State the problem:** Find the equation of the line passing through points $A(-5, 3)$ and $B(0, -7)$.\n\n2. **Formula used:** The slope $m$ of the line through points $(x_1, y_

1. **State the problem:** Evaluate the expression $$\frac{45}{b} + a$$ when $$a=5$$ and $$b=32$$, and simplify the expression $$4d + 3d$$. 2. **Substitute the values:** Replace $$a

1. **State the problem:** Evaluate the expression $$a^2 b^5 : c^{-10} + \frac{a - b}{a + b}$$ for $$a=2$$, $$b=4$$, and $$c=-\frac{1}{2}$$. 2. **Recall the rules:**

1. **State the problem:** Solve the equation $6 - 5k = 14 + 3k$ and find the value of $k$. 2. **Write the equation:**

1. **State the problem:** Find the point of intersection (POI) of the lines given by the equations $y = -2x + 6$ and $8x - 3y = -4$. 2. **Use substitution or elimination method:**

1. **State the problem:** Determine if $y$ is proportional to $x$ based on the given table values: $$\begin{array}{c|cccc} x & 3 & 5 & 7 & 9 \\ y & 24 & 45 & 70 & 99 \end{array}$$

1. **State the problem:** We need to find the unit price per padlock when a pack of 7 padlocks costs 23.80 dollars. 2. **Formula used:** The unit price is calculated by dividing th

1. **State the problem:** A potter makes bowls and planter pots. Each bowl requires 8 hours and 2 pounds of clay. Each planter pot requires 18 hours and 14 pounds of clay. The pott

1. Problem 14: Find functions $f$ and $g$ such that $h(x) = (f \circ g)(x)$ where $h(x) = \sqrt{x + 4}$. Step 1: Recall that $(f \circ g)(x) = f(g(x))$. We want to express $\sqrt{x

1. The user asks if a value is in the range 130-140. 2. To answer, we need to check if the value lies between 130 and 140 inclusive.

1. **State the problem:** We have a triangle with side lengths $2x$, $4x - 2$, and $2(x + 7)$ yards, and a square with side length $2.5x$ yards. The perimeter of the triangle equal