🧮 algebra

1. **State the problem:** We are given the equation $$6x + 2y = 120$$ representing the total capacity of a subway, where $$x$$ is seating capacity and $$y$$ is standing capacity. 2

1. **State the problem:** We are given the equation $$6x + 2y = 120$$ representing the total capacity of a subway, where $x$ is seating capacity and $y$ is standing capacity. 2. **

1. **State the problem:** Determine if the function $$h(x) = \frac{x}{x^2 - 1}$$ is even, odd, or neither. 2. **Recall definitions:**

1. **State the problem:** Determine whether the functions $$h(x)=3x^3+5$$, $$f(x)=x+|x|$$, and $$h(x)=\frac{x}{x^2}-1$$ are even, odd, or neither. 2. **Recall definitions:**

1. **State the problem:** We need to find the average rate of change of the function $$h(x)=x^2-2x+3$$ over the intervals $[-1,1]$, $[0,2]$, and $[2,5]$. 2. **Formula for average r

1. **State the problem:** We need to find the average rate of change of the function $$g(x)=x^3-2x+1$$ over three intervals: from $$-3$$ to $$-2$$, from $$1$$ to $$3$$, and from $$

1. **State the problem:** We need to find the average rate of change of the function $g(x)=x^3-2x+1$ over an interval $[a,b]$. 2. **Formula:** The average rate of change of a funct

1. **State the problem:** Solve the equation $$4x^2 - 9 = -409$$ for all real and complex values of $x$. 2. **Rewrite the equation:** Add $9$ to both sides to isolate the quadratic

1. **State the problem:** Solve the equation $$\frac{a - 6}{8} + \frac{8}{9} = \frac{a - 6}{16}$$. 2. **Identify the formula and rules:** This is a linear equation involving fracti

1. **State the problem:** We are given the function $$h(x) = 3x^3 + 5$$ and we want to understand or analyze it. 2. **Formula and rules:** This is a cubic function of the form $$h(

1. **State the problem:** Determine if the function $f(x)=4x^3$ is even, odd, or neither. 2. **Recall definitions:**

1. **State the problem:** Solve the inequality $$\lvert x+10 \rvert > 5$$. 2. **Recall the definition of absolute value inequality:** For $$\lvert A \rvert > B$$ where $$B > 0$$, t

1. **Problem statement:** Identify the fractions at $A$ and $B$ on the number line. 2. **Use the number line rule:**

1. **State the problem:** Solve for $n$ in the equation $$\frac{x^5 \cdot x^n}{x^8} = x^{20}.$$\n\n2. **Recall the exponent rules:** When multiplying powers with the same base, add

1. **State the problem:** Simplify the expression $$\frac{8w^4}{16v^2w^4 - 4wy^3}$$. 2. **Factor the denominator:**

1. **Problem statement:** Mrs Fisher has a 10% pay rise and her new wage is £22 per hour. We need to find her original wage before the rise. 2. **Understanding the problem:** A 10%

1. **State the problem:** Simplify the expression $a\sqrt{a^{-1}} \cdot \sqrt{a^{-1}} + \sqrt{a^{-1}}$. 2. **Recall the rules:**

1. **State the problem:** Find the multiplicative inverse of the complex number $5 - 12i$. 2. **Recall the formula:** The multiplicative inverse of a complex number $z = a + bi$ is

1. **State the problem:** Factorise the expression $$4x^2 - y^2$$. 2. **Recognize the formula:** This is a difference of squares, which follows the formula $$a^2 - b^2 = (a - b)(a

1. **State the problem:** Solve the inequality $$-4x + 2 < -6$$. 2. **Isolate the variable term:** Subtract 2 from both sides:

1. **State the problem:** Solve the equation $$\frac{4}{5} x + \frac{6}{7} = 6x + \frac{97}{3}$$ for $x$. 2. **Write down the equation:** $$\frac{4}{5} x + \frac{6}{7} = 6x + \frac