🧮 algebra

1. The problem is to find the next number in the sequence: 8, 2, \frac{1}{3}, \frac{1}{24}, ? 2. Let's analyze the pattern by looking at the relationship between consecutive terms.

1. **State the problem:** Solve for $x$ in the equation $$2(3x + 6 - x^2) = 7x + 5$$ to 3 significant figures. 2. **Expand and simplify:** Distribute the 2 on the left side:

1. The problem is to find the value of the missing number $?$ in the equation $1 + ? - ... = -1$ given the other rows and columns. 2. From the first row: $2 - 3 + 5 = 4$.

1. **State the problem:** We are given two equations: $$\frac{4a + 2}{11} = 2$$

1. **Problem Statement:** We are given that $y$ is inversely proportional to $x$, and when $x=2$, $y=\frac{1}{2}$. We need to find the graph that represents this relationship and t

1. **Stating the problem:** We know the selling price (SP) is 28.14, the profit or loss percentage based on cost price (CP) is 17.25%, and we need to find the cost price and the pr

1. The problem is to find the values of $y$ for given $x$ values in the exponential functions and fill the tables with the points $(x,y)$.\n\n2. The general form of an exponential

1. **State the problem:** We have the function $$f(x) = \frac{8x^3 - 27}{2x - 3}$$ which has a removable discontinuity at $$x = \frac{3}{2}$$. We want to find which function defini

1. **Find the equation of the circle given the center and radius or diameter.** The general equation of a circle with center $(h,k)$ and radius $r$ is:

1. The problem is to solve the equation: $$25x^2 = 16$$ 2. The formula used here is to isolate $x$ by taking the square root after dividing both sides by 25.

1. **Problem Statement:** Solve the equation $$10(x^4+1) + 63(x^3 - x) + 52x^2 = 0$$ 2. **Step 1: Rearrange and divide by $x^2$ (assuming $x \neq 0$):**

1. **State the problem:** Solve the inequality $|2x + 1| \geq 5$. 2. **Recall the definition of absolute value inequality:** For $|A| \geq B$ where $B \geq 0$, the solution is $A \

1. The problem is to identify and understand the numbered quadrants of a graph used for plotting linear equations. 2. The Cartesian coordinate plane is divided into four quadrants

1. **State the problem:** Find the equation of the line in slope-intercept form $y=mx+b$ that passes through the points $(-2,-7)$ and $(3,8)$.\n\n2. **Formula for slope:** The slop

1. **Problem Statement:** Define a residuated lattice and prove some initial properties. 2. **Definition:** A residuated lattice is an algebraic structure $(L, \wedge, \vee, \cdot,

1. **Problem:** Find $x$ such that $$(-5)^{x+1} \times (-5)^5 = (-5)^7.$$ 2. **Formula and rules:** When multiplying powers with the same base, add the exponents: $$a^m \times a^n

1. **State the problem:** Karl's truck starts with 400 liters of fuel and consumes 0.8 liters per kilometer driven. We want to find the relationship between the fuel remaining and

1. **State the problem:** We have a parabola defined by the equation $y = mx^2$ and two points on the curve: $(4, A)$ and $(8, B)$. The derivative is given by $\frac{dy}{dx} = 2mx$

1. The problem asks to express the transformed variables $X$ and $Y$ in terms of the original variables $x$ and $y$ from the given function $$y = \frac{2x}{x^3 - 8}.$$\n\n2. To lin

1. Let's start by stating the problem: What is a monic quadratic trinomial? 2. A quadratic trinomial is a polynomial of degree 2 with three terms, generally written as $$ax^2 + bx

1. **Problem Statement:** Fit a second degree polynomial of the form $$y = a + bx + cx^2$$ to the given data points: $$\begin{array}{c|ccccccc} X & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ y &