🧮 algebra

1. **Problem statement:** A networking technician has an initial annual salary of 40000. The salary increases by 2% of the initial salary every year. We need to find:

1. Let's start by understanding what indices (or exponents) are. An index tells us how many times to multiply a number by itself. For example, $a^n$ means multiply $a$ by itself $n

1. **State the problem:** Solve the exponential equation $$8 \cdot 0.25^{2x - 3} = 2^{x - 1}$$ for $x$. 2. **Rewrite the bases as powers of 2:** Note that $0.25 = \frac{1}{4} = 2^{

1. The problem is to simplify the expression without including the term $-\lg 100$. 2. Recall that $\lg$ denotes the base-10 logarithm, so $\lg 100 = 2$ because $100 = 10^2$.

1. **State the problem:** Simplify and analyze the expression $$\frac{a}{b} \cdot \frac{1}{x^2 + 2x - 8}$$. 2. **Identify the formula and rules:** We are multiplying a constant fra

1. **State the problem:** Simplify the expression $2 \lg 5 + 0.5 \lg 16 - \lg 100$. 2. **Recall logarithm properties:**

1. The problem is to graph a function and find its domain and range. 2. The domain of a function is the set of all possible input values (x-values) for which the function is define

1. **Problem statement:** A hard drive initially contains 1500 megabytes of data and loses 5 megabytes every hour due to corruption. 2. **Formula:** The amount of data left after $

1. **State the problem:** Solve the exponential equation $$2^{4x-9} = \left(\frac{1}{8}\right)^{x-4}$$. 2. **Rewrite the bases:** Note that $$\frac{1}{8} = 8^{-1}$$ and $$8 = 2^3$$

1. The problem is to simplify the expression $-350 + 420 - 180$. 2. We use the rule of addition and subtraction of integers: add positive numbers and subtract negative numbers step

1. **State the problem:** The temperature starts at $-12^\circ C$. It increases by $7^\circ C$ by noon and then decreases by $5^\circ C$ in the evening. We need to find the final t

1. **State the problem:** Solve the rational equation $$\frac{9}{x-7} - \frac{7}{x-6} = \frac{13}{x^2 - 13x + 42}$$. 2. **Recognize the denominator factorization:** The quadratic i

1. **State the problem:** Solve the rational equation. Since the user did not provide a specific equation, let's consider a general example: $$\frac{2x+3}{x-1} = \frac{4x-1}{x+2}$$

1. **State the problem:** Solve the equation $$\frac{9}{x-7} - \frac{7}{x-6} = \frac{13}{x^2 - 13x + 42}$$. 2. **Recognize the denominator factorization:** The quadratic in the den

1. **State the problem:** Solve the equation $$\frac{9}{x} - 7 - \frac{7}{x} - 6 = \frac{13}{x^2} - 13 + 42$$ for $x$. 2. **Rewrite and simplify the equation:** Combine like terms

1. **State the problem:** Solve the rational equation. Since the user did not specify a particular equation, let's consider a general example: $$\frac{2x+3}{x-1} = \frac{x+5}{x+2}$

1. **State the problem:** Solve the system of equations using the matrix inversion method: $$\begin{cases} x + 2y + 4z = 14 \\ 2x - y + 5z = 15 \\ -3x + 2y + 4z = 13 \end{cases}$$

1. **State the problem:** Simplify the expression $x \cdot 10 \sqrt{x^7}$. 2. **Rewrite the square root:** Recall that $\sqrt{x^7} = x^{7/2}$ because $\sqrt{x^a} = x^{a/2}$.

1. **State the problem:** Solve the system of linear equations: $$\begin{cases} x + 3y + 3z = 5 \\ 3x + y - 3z = 4 \\ -3x + 4y + 7z = -7 \end{cases}$$

1. **State the problem:** Find the x-intercept of the function $$y = 2 - 5x$$. 2. **Recall the definition:** The x-intercept is the point where the graph crosses the x-axis, which

1. **Problem vii:** Find the value of $h$ such that the vertex of $f(x) = (x + 4)(x - h)$ has a positive $x$-coordinate. 2. **Step 1:** Expand the function: