🧮 algebra

1. Let's consider a simple algebraic equation to solve: $$2x + 3 = 11$$. 2. The goal is to find the value of $x$ that makes this equation true.

1. **Stating the problem:** Simplify or manipulate the given algebraic expression. 2. **Formula and rules:** Use algebraic rules such as distributive property, combining like terms

1. Let's start by stating the problem clearly: We want to simplify the solution to make it easier to understand. 2. The key to simplification is breaking down the problem into smal

1. **State the problem:** Simplify the expression $$\frac{12}{1 - 9x^2} = \frac{1 - 3x}{1 + 3x} + \frac{1 + 3x}{3x - 1}$$ and understand how to work with these fractions. 2. **Reco

1. **Problem Statement:** Shade the regions corresponding to the fractions $\frac{4}{7}$, $\frac{13}{49}$, and $\frac{1}{6}$ in Fig. 2.1.

1. **State the problem:** We need to find the sign diagram for the quadratic function $$y = -x^2 + 4x - 3$$. 2. **Recall the formula and rules:** The sign diagram shows where the f

1. Énonçons le problème : Résoudre l'exercice 6 en détaillant chaque étape. 2. Comme vous n'avez pas précisé l'exercice 6, je vais supposer un problème algébrique classique : résou

1. The problem is to solve the exponential equation $$5^{x+3} = 5^{3x-1}$$ for $x$. 2. Since the bases on both sides are the same (base 5), we can set the exponents equal to each o

1. **Solve the equation:** $3(x + 4) = 5(x - 6) + 32$ Use the distributive property: $3x + 12 = 5x - 30 + 32$

1. **State the problem:** Simplify the expression $x^2 + x$. 2. **Formula and rules:** This is a polynomial expression. We can factor it by finding the greatest common factor (GCF)

1. **Problem statement:** (a) Given the equation $$18 (3 + \sqrt{c})(2\sqrt{c} - 3) = 1 + k\sqrt{c}$$ where $c$ and $k$ are prime numbers, find $c$ and $k$.

1. **Problem Statement:** We need to find the exponential function $f(x) = cb^x$ given two points on its graph: $(1, 6)$ and $(3, 24)$. 2. **Formula and Rules:** The general form o

1. **State the problem:** Simplify fully the expression $$\frac{6x^{3} + 13x^{2} - 5x}{4x^{2} - 25}$$

1. **Problem statement:** Solve the simultaneous equations graphically for two systems: (a) $2xy = 3$ and $x - 3y = -5$

1. **State the problem:** Find the values of $n$ such that $$\frac{10^{4n} \times 2^{3(n^2 - 5n)} \times 5^{2(1 - 2n)}}{20^2} = 1$$

1. **Problem 1:** Verify if $2017^2 = 2016^2 + 2016$. 2. Use the identity for the difference of squares:

1. نبدأ بكتابة الدالة المعطاة: $$h(x) = \frac{x + 1}{x - 1}$$ 2. المطلوب هو حساب التعبير: $$18 \times 2h(0) - h(-2)$$

1. **Problem:** Determine the answers for each of the given subproblems. 2. **Set notation and intervals:**

1. **State the problem:** We have a trapezium with parallel sides of lengths $x+6$ and $3x-4$, and the height is $x-1$. The area is given as 119 cm². 2. **Formula for the area of a

1. **State the problem:** We need to divide 0.072 by 0.00050. 2. **Recall the division rule:** Dividing two decimals can be simplified by converting the divisor to a whole number.

1. **State the problem:** We want to show algebraically that the decimal $0.306$ is equal to the fraction $\frac{34}{111}$. 2. **Express the decimal as a fraction:** Let $x = 0.306