🧮 algebra

1. **State the problem:** Solve the inequality $$\frac{12}{11} \geq -16u$$ for $u$. 2. **Recall the rule:** When multiplying or dividing both sides of an inequality by a negative n

1. **State the problem:** Solve the inequality $-12 < \frac{8}{3} y$ for $y$. 2. **Recall the rule:** When multiplying or dividing both sides of an inequality by a positive number,

1. **State the problem:** We are given two functions: $$g(x) = f(rx)$$ and $$f(x) = -3x + 5$$. We need to find the value of $$r$$ such that the graph of $$g(x)$$ matches the given

1. **State the problem:** Cory wants to buy a home theater costing $3800$. He pays $1000$ down and then makes $6$ equal interest-free payments. 2. **Find the remaining balance afte

1. **State the problem:** We have a parent function $f(x)$ which is a narrow parabola opening upwards with vertex at $(0,0)$.

1. **Problem Statement:** We are given two functions:

1. **State the problem:** We want to express the fraction $$\frac{6 + \sqrt{27}}{4 - \sqrt{3}}$$ in the form $$\frac{r + s\sqrt{3}}{13}$$ where $r$ and $s$ are integers. 2. **Simpl

1. **State the problem:** Rationalise the denominator of the expression $$\frac{(6 + \sqrt{3})(6 - \sqrt{3})}{\sqrt{33}}$$. 2. **Recall the formula:** To rationalise a denominator

1. **State the problem:** We need to find the x-intercepts of the given parabola. 2. **Recall the definition:** The x-intercepts are the points where the graph crosses the x-axis,

1. **Problem Statement:** Identify the translation parameters $h$ and $k$ for each pair of parent function $f(x)$ and translated function $g(x)$, then write the equation of $g(x)$.

1. **State the problem:** We need to find the average speed of the trip there, given the return trip speed and times for both trips.

1. **State the problem:** Chelsea drives away from the hardware store at $70$ km/h. Stephanie starts $4$ hours later, driving in the opposite direction at $65$ km/h. We want to fin

1. **State the problem:** We are given two functions: $$h(x) = x^2 - 4$$

1. **State the problem:** We are given two functions: $$h(x) = x^2 + 3$$

1. **State the problem:** Determine if the expression $$y=\frac{3x+4}{x-2}$$ represents a function. 2. **Recall the definition of a function:** A function assigns exactly one outpu

1. **State the problem:** Solve the equation $3y^2 - 20 = 160 - 2y^2$ for $y$. 2. **Write down the formula and rules:** This is a quadratic equation. To solve, first collect like t

1. **State the problem:** We need to determine if the expression $$y=\frac{3x+4}{x-2}$$ defines a function. 2. **Recall the definition of a function:** A function assigns exactly o

1. **State the problem:** We need to find the value of $\ln \sqrt[5]{e^2}$. 2. **Recall the properties of logarithms and exponents:**

1. **State the problem:** We need to find the value of $\ln e^5$. 2. **Recall the formula:** The natural logarithm function $\ln(x)$ is the inverse of the exponential function $e^x

1. **State the problem:** Solve the equation $$ (2x-3)^2 - \frac{4}{(2x-3)^2} - 3 = 0 $$ using a suitable substitution.

1. **State the problem:** We need to find a number that is $21.5\%$ smaller than $62$. 2. **Formula:** To find a percentage decrease, use the formula: