🧮 algebra

1. Let's start by understanding the **Real-Number System**. It includes all rational and irrational numbers. Rational numbers can be expressed as fractions, while irrational number

1. Problem 8: Calculate Jade's total yearly income. Jade's income consists of three parts: annual salary, commission on sales, and car allowance.

1. Il problema chiede di risolvere la disequazione $$\log_2(x) - \log_2(8) > 0$$ nell'insieme dei numeri reali. 2. Ricordiamo la proprietà dei logaritmi: $$\log_a(b) - \log_a(c) =

1. Il problema chiede di risolvere la disequazione $$\log_2(x) - \log_2(8) > 0$$ nell'insieme dei numeri reali. 2. Ricordiamo la proprietà dei logaritmi: $$\log_a(b) - \log_a(c) =

1. **State the problem:** We have points A(0,8) and B(16,0). Point D divides segment AB in ratio 1:3. Line L passes through D with gradient $\sqrt{3}$ and also passes through $(-2,

1. Problem statement: Löse die Division von Brüchen mit Hilfe des Kehrwerts und kürze wenn möglich. 2. Formel: \( \frac{a}{b} : \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} \)

1. **State the problem:** Solve the equation $37 \times 1 - \frac{1}{x} = \frac{6}{9}$ for $x$. 2. **Rewrite the equation:** Since $37 \times 1 = 37$, the equation becomes:

1. **State the problem:** Solve the equation $$\ln\left(\frac{x^2+4}{4}\right) = \ln 4$$ for $x$. 2. **Recall the property of logarithms:** If $\ln A = \ln B$, then $A = B$, provid

1. **State the problem:** Simplify the expression $$\left(4a^{-\frac{4}{5}} b^{\frac{1}{10}}\right)^{\frac{5}{2}} \left(a^{\frac{1}{3}} b^{\frac{1}{4}}\right)^3$$. 2. **Recall the

1. **State the problem:** Simplify the expression $$\left(\frac{a^3 b^{-2}}{a^{-1} b^{5/2}}\right)^2$$. 2. **Recall the rules:**

1. **State the problem:** Solve the linear equation $$\frac{1}{2} - \frac{1}{3}x = 1$$. 2. **Rewrite the equation:** The equation is $$\frac{1}{2} - \frac{1}{3}x = 1$$.

1. **Problem:** Factor the trinomial $x^2 + 12x + 36$ completely. 2. **Formula and rules:** A trinomial of the form $x^2 + bx + c$ can be factored as $(x + m)(x + n)$ where $m$ and

1. The problem is to analyze the function $f(x) = \log_2(x) + 2$. 2. The logarithmic function $\log_2(x)$ is defined only for $x > 0$.

1. **State the problem:** Simplify the expression $$\frac{5}{h} - 2h^2 - 6h + 15f$$ where $h$ and $f$ are variables. 2. **Understand the terms:** The expression contains a rational

1. **State the problem:** Factor the quadratic expression $x^2 + 16x - 36$. 2. **Recall the factoring formula:** For a quadratic $ax^2 + bx + c$, we look for two numbers that multi

1. The problem asks to find the value of $x$ for which $f(x) = x^2 + x + 1$ equals the given points in the set $\{(-2,8), (-1,0), (1,1), (m,-1)\}$. 2. We use the function $f(x) = x

1. **State the problem:** Simplify the expression $4x - 2x^2 - 2$ or rewrite it in a standard polynomial form. 2. **Rewrite the expression:** The given expression is $4x - 2x^2 - 2

1. **State the problem:** Solve the equation $$\frac{2}{3}x + 5 = 4 - \frac{3}{4}x$$ for $x$. 2. **Write down the equation:** $$\frac{2}{3}x + 5 = 4 - \frac{3}{4}x$$

1. **Stating the problem:** We are given that Emma is $y$ years old, Hanna is 2 years older than Emma, and together their ages sum to 26 years. We need to write an equation for the

1. **State the problem:** Find the difference quotient $$\frac{f(x+h)-f(x)}{h}$$ for the function $$f(x) = x^2 - 3x$$.

1. The problem asks to express each number as a power with exponent $\frac{1}{2}$ and then write the answer as a radical. 2. Recall that raising a number to the power $\frac{1}{2}$