🧮 algebra

1. **State the problem:** We are given the function $f(x) = 4x - \frac{1}{x}$ and asked to find $f(x + \Delta x)$. 2. **Recall the formula:** To find $f(x + \Delta x)$, substitute

1. **State the problem:** Solve the inequality $$-2x^2 + 9x - 17 > -8x - 9$$ algebraically. 2. **Bring all terms to one side:** Add $$8x + 9$$ to both sides to get

1. **State the problem:** Solve the inequality $$-2x^2 + 12x + 52 \leq -2$$ algebraically. 2. **Rewrite the inequality:** Move all terms to one side to set the inequality to zero:

1. **Stating the problem:** Solve for the missing number in the equation $15 - \_ = 4$. 2. **Formula and rules:** To find the missing number, isolate it by performing inverse opera

1. **State the problem:** Find the value of $y$ when given the expression $2(-4) + 3 + \frac{8}{-4}$. 2. **Write the expression:**

1. **State the problem:** Solve the equation $x^{-1} = -0.25$ for $x$. 2. **Recall the meaning of negative exponents:** $x^{-1} = \frac{1}{x}$.

1. **State the problem:** Solve the linear equation $y - 2y = 4x + 6$ for $y$. 2. **Simplify the left side:** Combine like terms on the left side.

1. The problem asks to evaluate $4^\circ$, which means $4$ degrees. 2. Degrees are a unit of angle measurement, not a number to be raised to a power.

1. **State the problem:** Solve the equation $3 = 3$. 2. **Analyze the equation:** The equation $3 = 3$ is a statement that both sides are equal.

1. **State the problem:** 17 more than a number is 43. Find the number. 2. **Choose the correct equation:** The problem translates to the equation $$17 + n = 43$$ where $n$ is the

1. **State the problem:** Solve the logarithmic equation $$\log_2 (x - 2) + \log_2 (x - 3) = 0$$ for $$x$$. 2. **Use logarithm properties:** Recall the product rule for logarithms:

1. **State the problem:** We are given the hyperbola equation $$\frac{(y - 2)^2}{25} - \frac{x^2}{16} = 1$$ and need to understand its properties and graph it. 2. **Identify the ty

1. **Plantear el problema:** Dada la ecuación de la elipse $$9x^2 + y^2 - 54x + 6y + 54 = 0$$, debemos reescribirla en su forma estándar para identificar su centro, ejes y radios.

1. **State the problem:** We are given the equation $$16x^2 + y^2 = 16$$ and asked to analyze it. 2. **Identify the type of conic:** This is an ellipse equation in standard form. T

1. **State the problem:** We are given the equation of an ellipse: $$\frac{(x+1)^2}{9} + \frac{(y+3)^2}{4} = 1$$

1. **State the problem:** Evaluate the expression $4^{-5} \cdot 4^{5}$. 2. **Recall the exponent multiplication rule:** When multiplying powers with the same base, add the exponent

1. **State the problem:** Evaluate the expression $$(-9)^2 \cdot (-9)^2$$. 2. **Recall the formula and rules:** When raising a number to a power, $$a^n$$ means multiplying $$a$$ by

1. **State the problem:** We need to express the equation $8\sqrt{a} = a^n$ in terms of $n$. 2. **Recall the rules:** The square root of $a$ can be written as a fractional exponent

1. **State the problem:** Given the expression $$\sqrt[5]{b^7} = b^m$$, find the value of the exponent $$m$$. 2. **Raise both sides to the power of 5:**

1. **State the problem:** Solve the equation $$a^3 + a^2 = 36$$ for the variable $a$. 2. **Rewrite the equation:** We want to find $a$ such that $$a^3 + a^2 - 36 = 0$$.

1. El problema nos pide identificar la expresión algebraica que representa la función exponencial mostrada en la gráfica. 2. La gráfica muestra una curva exponencial que crece rápi