🧮 algebra

1. **State the problem:** Solve the equation $2^x = 0.25$ for $x$. 2. **Recall the formula and rules:** We know that $0.25$ can be written as a power of 2 because $0.25 = \frac{1}{

1. **State the problem:** Solve the equation $2^x = 0.25$ for $x$. 2. **Recall the formula and rules:** We know that $0.25$ can be written as a power of 2 because $0.25 = \frac{1}{

1. **Problem Statement:** We are given a graph that starts near $y=6$ at $x=0$, decreases sharply to $y=0$ near $x=1$, rises to a peak near $y=2.5$ at $x=2$, then falls again to $y

1. The problem is to identify the function represented by the given graph. 2. The graph starts at approximately (0, 6), decreases steeply to (1, 0), rises to a peak near (2, 2.5),

1. Solve $6 = \frac{a}{4} + 2$. Subtract 2 from both sides: $6 - 2 = \frac{a}{4}$.

1. **State the problem:** We need to find the 28th term $a_{28}$ of the sequence defined by the recurrence relation $a_n = a_{n-1} + \frac{5}{2}$, given that the 26th term $a_{26}

1. **Problem:** Given the height of a tree $h = 1.7t + 28$, find the slope and y-intercept and explain their meanings. 2. **Formula:** The equation of a line is $y = mt + b$, where

1. Given the slope $m=\frac{2}{3}$ and y-intercept $b=-10$, the line equation is $y=\frac{2}{3}x-10$. 2. Given slope $m=-\frac{7}{9}$ and point $(0,-\frac{1}{2})$, since $x=0$ is t

1. The problem asks us to transform the graph of the function $y=f(x)$ (dashed curve) to the graph of $y=-f(x-3)-3$ (solid curve). 2. The transformation involves three steps applie

1. Evaluate without using a calculator: (a) Given $f(x) = 4^x$, find $f(0.5)$, $f(\sqrt{2})$, $f(\pi)$, $f(-\frac{1}{3})$.

1. Let's clarify the problem: You want to move a point or a graph 3 units to the right and 4 units down. 2. The general rule for moving a graph or point horizontally and vertically

1. The problem asks us to transform the graph of the function $y=f(x)$ (dashed curve) to the graph of $y=f(-x+3)-4$ (solid curve). 2. The transformation involves two steps inside t

1. The problem asks us to transform the graph of $y=f(x)$ (dashed curve) to the graph of $y=-f(x-4)-2$ (solid curve). 2. The transformation involves three steps based on the functi

1. **State the problem:** We are given the manufacturer's suggested retail price (MSRP) of a car as 24880 and its expected value after 4 years as 13924. We need to find a linear de

1. **State the problem:** We need to find a linear depreciation function $f(x)$ for a car whose value decreases from 25280 to 11765 over 5 years. 2. **Formula and rules:** A linear

1. Let's clarify the problem: we are solving a quadratic equation that yields the solutions -0.8 and 4.8. 2. The general form of a quadratic equation is $$ax^2 + bx + c = 0$$.

1. **Problem (a):** Given the equations $$4 \cdot 36 = 100\alpha$$

1. **State the problem:** We need to solve the simultaneous equations graphically: $$y = x^2 - x - 6$$

1. **Problem statement:** (a) Given the expressions involving infinite decimals and ratios, simplify and find the value of $x$ in the equation:

1. **State the problem:** We need to find the value of $x$ such that the ratio $\frac{x - 6}{2x}$ is equivalent to the ratio $\frac{1}{3}$. 2. **Write the equation from the ratio:*

1. Problem 200: Prove that $$A = \sqrt[3]{(a - 3b) \sqrt{a^2 - 9b^2}} \cdot \sqrt[3]{(a + 3b) \sqrt{a^2 - 9b^2}} = 1$$