🧮 algebra

1. **State the problem:** We have two numbers, say $a$ and $b$. One number is increased by $6$, so it becomes $a + 6$. The other is decreased by $2$, so it becomes $b - 2$.

1. **State the problem:** We need to find the value of $96 \times 104$ using an algebraic identity. 2. **Formula used:** We use the identity for the product of two numbers close to

1. **State the problem:** Given that $FG=KL$ and $QP=ST$, prove that $FG - QP = KL - ST$.

1. **State the problem:** We need to find the values of the function $$f(x) = \frac{1}{2}x$$ at $$x=0$$, $$x=2$$, and $$x=-2$$. 2. **Formula:** The function is given by $$f(x) = \f

1. **State the problem:** We are given a graph of a vertical line at $x = -4$ and asked to find the domain and range of the relation, and to determine if it is a function using the

1. **State the problem:** We need to graph the piecewise function: $$f(x)=\begin{cases}-\frac{2}{3}x+1, & x<3 \\ -1, & x\geq 3\end{cases}$$

1. Vamos começar declarando o problema: temos duas variáveis $a$ e $b$ tais que $a+b=40$ e $ab=72$. Queremos encontrar o valor de $a-b$. 2. Para resolver isso, usamos a fórmula da

1. **Problem c:** Solve the system of inequalities: $$ (2x + 1)^2 - 9 \leq 0 \quad \text{and} \quad 2x < 6 $$

1. **State the problem:** We need to complete the table for values of $x$ and corresponding values of $10^x$, then plot these points on a graph with a logarithmic y-axis. 2. **Reca

1. **State the problem:** We need to determine the range of the graph described, which is the set of all possible $y$-values the function takes. 2. **Analyze the graph description:

1. The problem asks for the domain of the function defined by the given table. 2. The domain of a function is the set of all possible input values (x-values) for which the function

1. The problem is to graph the solutions $x_1 = -2$ and $x_2 = -3$ on a number line. 2. These values are the roots of a quadratic equation, and we want to show their positions visu

1. Problemet handlar om att avgöra om $\frac{13}{4}$ är det enda svaret. 2. Om detta är en lösning till en ekvation eller ett uttryck, måste vi undersöka om det finns fler lösninga

1. **State the problem:** Solve the equation $$\frac{1}{x-3} = 4$$ for $x$. 2. **Use the formula:** To solve for $x$, multiply both sides by the denominator to eliminate the fracti

1. **State the problem:** Solve the quadratic equation $$x^2 + 5x + 6 = 0$$ using the quadratic formula and discriminant. 2. **Recall the quadratic formula:** For an equation $$ax^

1. **State the problem:** Divide the polynomial $$33x^5 + 22x^4 - 50x^3 - 26x^2 + 13x$$ by $$3x^2 + 2x - 1$$ using long division. 2. **Recall the formula and rules:** Polynomial lo

1. **State the problem:** Divide the polynomial $$-30x^6 + 48x^5 + 50x - 80$$ by the binomial $$5x - 8$$ using long division. 2. **Recall the formula and rules:** Polynomial long d

1. **State the problem:** Find the roots of the quadratic equation $$5x^2 - 8x + 1 = 0$$. 2. **Formula used:** The roots of a quadratic equation $$ax^2 + bx + c = 0$$ are given by

1. **State the problem:** We need to find $k^{-1}(-1)$ where $k(x) = f(x^3)$. 2. **Understand the inverse function:** $k^{-1}(-1)$ means the value of $x$ such that $k(x) = -1$.

1. **State the problem:** Solve the equation $$5(-8+x)=6x+\frac{1}{5}$$ for $$x$$. 2. **Write the equation:** $$5(-8+x)=6x+\frac{1}{5}$$.

1. **State the problem:** Simplify the expression $$\frac{p^{-5}}{7^{-2}q^{-9}}$$ and write the answer using only positive exponents. 2. **Recall the exponent rules:**