🧮 algebra

1. **State the problem:** Simplify the rational expression $$\frac{x^2 - 7x - 18}{2x} \div (x - 9)$$. 2. **Rewrite the division as multiplication:** Dividing by a quantity is the s

1. **State the problem:** Solve the equation $$\frac{2x+4}{3} = 5$$ for $x$. 2. **Formula and rules:** To solve for $x$ in a fraction equation, multiply both sides by the denominat

1. The problem is to find the range of a function, which means determining all possible output values (y-values) the function can produce. 2. To find the range, we often start by u

1. **State the problem:** We are given the linear function $f(x) = -\frac{4}{3}x - 1$ and asked to find its domain. 2. **Recall the domain of linear functions:** The domain of a li

1. **State the problem:** Solve the inequality $$x^2 - 3x > \frac{1}{2}x^2 - 4$$. 2. **Rewrite the inequality:** Move all terms to one side to compare with zero.

1. **State the problem:** We are given the function $f(x) = 5x + 6$ and asked to find the value of $x$ when $f(x) = 46$. 2. **Write the equation:** Set the function equal to 46:

1. Stating the problem: We need to find the value of $x$ such that $f(x) = 19$ for the function $f(x) = 2x + 3$. 2. Write the equation:

1. **State the problem:** Solve the equation $$\sqrt{5x - 1} + 4 = 11$$ for $x$. 2. **Isolate the square root term:** Subtract 4 from both sides to get $$\sqrt{5x - 1} = 11 - 4$$

1. **State the problem:** Solve the equation and check for extraneous solutions. 2. **General approach:** When solving equations, especially those involving radicals, fractions, or

1. Problema 605: Equazione $x^2 + 2kx - (1+k^2) = 0$. a. Radici antireciproche: il prodotto delle radici $p = -1$.

1. The problem is to solve the equation using a different method than previously used. 2. Let's assume the equation is $ax+b=0$ (since the user did not specify, we solve a general

1. **State the problem:** We are given the equation

1. The problem is to graph the line with slope 3 and y-intercept 2. 2. The formula for a line in slope-intercept form is $$y = mx + b$$ where $m$ is the slope and $b$ is the y-inte

1. **State the problem:** Solve the system of linear equations: $$\begin{cases} x + 2y + z = 0 \\ 2x + y + z = 1 \\ x + y + 2z = 1 \end{cases}$$

1. Let's clarify the problem: You are asking why an arbitrary value on the x-axis appears at a seemingly random point rather than a specific stated part. 2. In mathematics, the x-a

1. **State the problem:** Express the rational function $$\frac{2x^3 - 4}{(x+3)(x-1)}$$ in partial fractions. 2. **Understand the problem:** Partial fraction decomposition requires

1. The denominator of a fraction is the part below the fraction line. 2. A denominator cannot be zero because division by zero is undefined in mathematics.

1. The problem is to solve the inequality $\left| \frac{x-4}{3x+1} \right| \geq 0$. 2. Recall that the absolute value of any real number or expression is always greater than or equ

1. The problem is to solve the inequality $$\left|\frac{x+4}{3x+1}\right| \geq 0$$. 2. Recall that the absolute value of any real number or expression is always greater than or equ

1. **State the problem:** Simplify the expression $\frac{3}{2}x - x$. 2. **Recall the rule:** When subtracting like terms, subtract their coefficients and keep the variable the sam

1. The problem is to solve the inequality $|x^4 + 2x^2 + 1| < 0$. 2. Recall that the absolute value $|A|$ of any expression $A$ is always greater than or equal to zero, i.e., $|A|