🧮 algebra

1. **Stating the problem:** We want to understand what "standard form" means in algebra and see examples. 2. **Definition:** In algebra, the standard form of a linear equation is u

1. **State the problem:** Find and simplify the expression for $$f(g(h(x)))$$ given $$f(x) = x + 1$$, $$g(x) = x^2$$, and $$h(x) = \sqrt{x}$$. 2. **Evaluate the innermost function:

1. **Énoncé du problème** : On considère la suite $(u_n)_{n\in\mathbb{N}}$ définie par : $$u_0 = 3, \quad u_{n+1} = (a+1)(u_n + 2)$$

1. **State the problem:** We want to find the inequalities that define the region $R$ bounded by the lines $x=7$, $y=2$, and $y=x$. 2. **Analyze each boundary:**

1. The problem is to find the asymptotes of a function, but the specific function is not provided. 2. Asymptotes are lines that a graph approaches but never touches.

1. The problem is to express and understand the expression "2 times 3 to the power of x". 2. This expression can be written mathematically as $$2 \times 3^x$$.

1. The problem states that the diagonal line is given by the equation $y = x$. 2. The region described is the set of points where $y \geq x$, meaning all points on or above the lin

1. The problem is to find the inequalities that define the region $R$ bounded by the lines $x=6$, $y=x$, and $y=1$. 2. First, note the lines:

1. **State the problem:** Fully factorise the expression $$7x - 18x - 6 + 10x^2$$. 2. **Simplify the expression:** Combine like terms for the $x$ terms:

1. Solve $33x^2 - 16 \leq 0$: - Rewrite as $33x^2 \leq 16$.

1. Solve $x^2 - 16 \leq 0$: - Rewrite as $(x-4)(x+4) \leq 0$.

1. **Understanding Exponential Functions:** An exponential function has the form $$y = a \cdot b^x$$ where $a \neq 0$, $b > 0$, and $b \neq 1$. The base $b$ determines the growth o

1. **State the problem:** Rekha and Dema both work in sales. Rekha earns a salary of 12000 per month plus a 5% commission on sales. Dema earns a salary of 12500 per month plus a 3%

1. **Montrer par récurrence que $S_n = \sum_{k=1}^n k(k+1) = \frac{n(n+1)(n+2)}{3}$**. 2. **Initialisation**: Pour $n=1$, $S_1 = 1 \times 2 = 2$.

1. Diberikan sistem persamaan linear 3 variabel: $$\begin{cases} 2x + y - z = 8 \\ -3x - y + 2z = -11 \\ -2x + y + 2z = -3 \end{cases}$$

1. **State the problem:** Jill starts with a full pail containing 2 gallons of water. On her way back, she loses half a quart of water. We need to find out how much water she manag

1. **State the problem:** A motorist travels from town X to town Z with stops and varying speeds, starting at 7:00am. We need to analyze the journey and represent it on a graph. 2.

1. **State the problem:** There are 55 musicians playing either oboe, double bass, or trombone. Each piece is in a major or minor key. We know 3/5 of the pieces are minor, and 7 mu

1. **State the problem:** There are 55 musicians playing pieces on oboe, double bass, or trombone. Each piece is either in a major or minor key. 2. **Given data:**

1. The problem states: 45% of $k$ is 72. 2. We write this as an equation: $0.45 \times k = 72$.

1. The problem asks to find what percentage 510 kg is less than 750 kg. 2. To find the percentage decrease, use the formula: