🧮 algebra

1. **State the problem:** We are given two functions: $$f(x) = \frac{x-2}{x+2}$$

1. **State the problem:** Pedro creates patterns of dominoes with dots. Each pattern has a certain number of dominoes and dots. We want to find which pattern number corresponds to

1. The problem is to locate the vertical asymptotes of a function. 2. Vertical asymptotes occur where the function's denominator is zero and the numerator is not zero at those poin

1. **State the problem:** Solve the equation $$0 = \left(\frac{1}{2}\right)^x$$ for $x$. 2. **Recall the properties of exponential functions:** For any base $a > 0$ and $a \neq 1$,

1. **State the problem:** Solve the equation $$\sqrt{x} + 2 = \sqrt{x} + 10$$. 2. **Analyze the equation:** We have the same term $$\sqrt{x}$$ on both sides of the equation.

1. **Problem statement:** Solve the equation $$ (6)(x-3)5x = 0 $$ for $x$. 2. **Formula and rules:** The product of factors equals zero if and only if at least one of the factors i

1. **State the problem:** We need to find the value of $x$ such that $f(x) = 4$ given the function $f(x) = 2x + 2$. 2. **Write the equation:** Set $f(x)$ equal to 4:

1. The problem is to find a quadratic equation with integer coefficients given the solutions $x = -5$ and $x = 3$. 2. The formula for a quadratic equation with roots $r_1$ and $r_2

1. **State the problem:** Simplify the expression $\frac{1}{2} \div \frac{1}{4} + \frac{3}{4} + \frac{4}{5}$.\n\n2. **Recall the division rule for fractions:** Dividing by a fracti

1. **State the problem:** Simplify the expression $$\frac{1}{4} \times -\frac{3}{5} + \frac{2}{5} \div -\frac{1}{2}$$. 2. **Recall the rules:**

1. **State the problem:** Solve the equation $$\log_{2x}(3x^2 + 10x - 16) = 2$$ for all values of $x$. 2. **Recall the logarithm definition:** If $$\log_a b = c$$, then $$a^c = b$$

1. **State the problem:** Solve the logarithmic equation $$\log_{x+1}(7x - 3) = 2$$ for $x$. 2. **Recall the logarithm definition:** If $$\log_a b = c$$, then $$a^c = b$$.

1. The problem asks to find quadratic equations with integer coefficients given the solutions. 2. Recall that if $x=r_1$ and $x=r_2$ are roots, the quadratic equation is given by:

1. The problem asks which number line shows points at every location representing a number with an absolute value of 4.5. 2. Recall the definition of absolute value: for any number

1. **State the problem:** Solve for $v$ in the equation $$40 = \frac{8}{5} v$$. 2. **Formula and rules:** To isolate $v$, divide both sides of the equation by the coefficient of $v

1. The problem is to solve for $y$ in the equation: $$20.48 = 8y$$

1. **State the problem:** Solve for $4x$ given the equation $2x + 15 = 7(4x - 9)$. 2. **Write the equation:**

1. The problem asks for the x-intercept of the line given by the equation $$4x - 5y = 2$$ in the xy-plane. The x-intercept is the point where the graph crosses the x-axis, so the y

1. **State the problem:** We need to find an equation that represents the relationship between $x$ and $y$ given the table: $$\begin{array}{c|cccc}

1. The problem is to find the factors of 12. 2. Factors of a number are integers that divide the number exactly without leaving a remainder.

1. The problem asks for the range of a parabola that opens upwards with vertex at approximately $(2,1)$. 2. The vertex form of a parabola is given by $$y = a(x-h)^2 + k$$ where $(h