🧮 algebra

1. **Problem:** Write the inequality $x < 4$ in interval notation and graph it. 2. **Formula and rules:**

1. **State the problem:** We are given a population function $$f(x) = x^3 - x^2 - 4x + 4$$ where $$x$$ is the number of years from now. We need to find the population 8 years from

1. **Stating the problem:** We have two sequences:

1. The problem is to create a wave function suitable for graphing on Desmos. 2. A common wave function is the sine function, which oscillates smoothly and periodically.

1. The problem is to simplify the expression $5^{3125^{3125}}$. 2. This is an exponential expression where the exponent itself is a power: $5^{(3125^{3125})}$.

1. **Stating the problem:** Evaluate the expression $5^^^5$, where $^^^$ denotes the tetration operation (repeated exponentiation).

1. The problem is to simplify or understand the expression $\frac{1}{x} - 1$. 2. The expression consists of two terms: $\frac{1}{x}$ and $-1$.

1. **State the problem:** Simplify the expression $$\frac{1}{x+1} + \frac{2}{(1+x)^2} - \frac{4}{(1-x)^4}$$. 2. **Rewrite the expression:** Note that $$x+1 = 1+x$$, so the first tw

1. **Problem statement:** Solve the quadratic equation without using the quadratic formula. 2. **General form:** A quadratic equation is generally written as $$ax^2 + bx + c = 0$$

1. The problem is to solve the equation given previously. Since no specific equation was provided, I will demonstrate solving a common algebraic equation example: $2x + 3 = 7$. 2.

1. **State the problem:** We need to find the coordinates of the midpoint of points $A$ and $B$, where $A$ and $B$ are the intersection points of the curve $y = (10x - 3)(x+1)$ and

1. The problem is to understand when a product or sum of numbers is positive or negative. 2. For multiplication: The product of two numbers is positive if both numbers have the sam

1. **State the problem:** Simplify the expression $$2x\left(2(\ln(x)^2)-3\ln(x)+2\right)-\frac{7x}{4}$$. 2. **Recall the distributive property:** To simplify, distribute $2x$ acros

1. The problem states: If $a * b$ means $\frac{1}{4}(a - b)$, find the value of $5 * 3$. 2. The operation $*$ is defined as $a * b = \frac{1}{4}(a - b)$.

1. **Stating the problem:** Simplify the expression $$x + 11 + (1 + x)^{22} - (1 - x)^{44}$$. 2. **Understanding the expression:** The expression contains terms with powers: $(1 +

1. **State the problem:** We are given a custom operation defined as $x * y = 3x - 2y$. 2. **Find the value of $2 * 5$ using the given operation:**

1. The problem is to analyze the set $\{2,4,6,8,10\}$. 2. This set contains five even numbers, each increasing by 2.

1. The problem states that the operation $a * b$ is defined as $2a + b$. 2. We need to find the value of $3 * 1$ using this definition.

1. **State the problem:** We need to solve the simultaneous equations:

1. The problem asks how to find $z_k$ in functions $f$ and $g$, and why $z_k$ appears in $e$. 2. Typically, $z_k$ represents a specific value or root related to the function, often

1. **Problem statement:** Given functions $f(x) = x^2 + 1$ and $g(x) = \sqrt{x - 4}$, find the composite functions $g(f(x))$ and $f(g(x))$ and their domains. 2. **Recall:** The com