🧮 algebra

1. **State the problem:** Solve the quadratic equation $3x^2 - 7x = 20$ for $x$. 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero:

1. The problem is to solve for $x$ in the equation $\log_x \left( \frac{4}{9} \right) = -2$. 2. Recall the definition of logarithm: $\log_a b = c$ means $a^c = b$.

1. El problema no está explícito, pero resolveré una ecuación común para ilustrar. 2. Supongamos que la ecuación es $$2x + 3 = 7$$.

1. **State the problem:** Solve the inequality $$\left|\frac{x+9}{x-9}\right| \le 2.$$\n\n2. **Rewrite the absolute value inequality:** This means \(-2 \le \frac{x+9}{x-9} \le 2\).

1. **State the problem:** Solve the equation $$\frac{2x+3}{4} = \frac{x-1}{2}$$ for $x$. 2. **Formula and rules:** To solve equations with fractions, multiply both sides by the lea

1. **State the problem:** Find the inverse function of $f(x) = \frac{1}{x+5}$ where $x \neq -5$. 2. **Recall the formula for inverse functions:** To find the inverse, swap $x$ and

1. **State the problem:** Find the equation of a line passing through the point $(2,8)$ and parallel to the line segment $GH$. 2. **Identify the slope of line $GH$:** The line $GH$

1. **State the problem:** Find the equation of a line passing through the point $(2,8)$ and parallel to the line segment $GH$. 2. **Identify the slope of line $GH$:** The line $GH$

1. **State the problem:** Find the intersection point of the two lines given by the equations: $$2x = y + 5$$

1. The problem asks to find an expression equivalent to $x^2 - 81$. 2. Recognize that $x^2 - 81$ is a difference of squares, which follows the formula:

1. **State the problem:** Solve the system of equations: $$8x + 4y = 12$$

1. **State the problem:** Convert the mixed number $5 \frac{4}{6}$ to an improper fraction. 2. **Formula:** To convert a mixed number $a \frac{b}{c}$ to an improper fraction, use:

1. Problem: Find $x$ in the equation $x^x = n^n$. 2. Formula and rules: When bases and exponents are equal, the expressions are equal. Here, $x^x = n^n$ implies $x = n$ because the

1. **State the problem:** Find the real solution(s) to the quadratic equation $$4x^2 + 8x + 3 = 0$$. 2. **Recall the quadratic formula:** For an equation $$ax^2 + bx + c = 0$$, the

1. El problema nos pide evaluar la expresión algebraica $$2x - 2x - 2$$ cuando $$x = 3$$. 2. La expresión es $$2x - 2x - 2$$. Primero, recordemos que al sustituir $$x$$ por un núme

1. **State the problem:** Simplify the expression $6a - [3 - (5 - 3a)] - 4$. 2. **Recall the rules:** When simplifying expressions with brackets, start from the innermost brackets

1. **State the problem:** Simplify the expression $$\frac{2xy}{x} \div \frac{6xy}{x^2}$$. 2. **Rewrite the division as multiplication by the reciprocal:**

1. **State the problem:** Solve the equation $$r = x - y$$ for the variable $$x$$. 2. **Formula and rules:** To isolate $$x$$, we need to undo the subtraction of $$y$$ by adding $$

1. **State the problem:** Solve for $y$ in the equation $y - 18 = c$. 2. **Formula and rules:** To isolate $y$, add 18 to both sides of the equation. This uses the addition propert

1. **State the problem:** Solve the linear equation $$5\left(\frac{2}{9}x + \frac{1}{3}\right) + \frac{3}{2} = \frac{1}{6}x$$. 2. **Distribute the 5:**

1. **State the problem:** Solve the equation $$5(0.2x + \frac{1}{3}) + \frac{3}{2} = \frac{1}{6}x$$ for $x$. 2. **Distribute and simplify:** Multiply 5 by each term inside the pare