Simplify the expression: $3\left(2x - \frac{1}{3}\right) - 5x + 1$.
Which property justifies the step: $4 + (x + 3) = 4 + (3 + x)$?
Solve for $x$: $\frac{2}{3}(6x - 9) = \frac{1}{2}(4x + 10)$.
Find the solution set for the inequality: $-4(x - 3) + 2x \ge 5x - 2$.
Which ordered pair is NOT a solution to the inequality $2x - 3y < 6$?
What is the slope of a line perpendicular to the line defined by $3x - 5y = 15$?
A line passes through $(-2, 5)$ and is parallel to $y = -\frac{1}{4}x + 7$. Write the equation of this line in Standard Form ($Ax + By = C$, where $A, B, C$ are integers and $A > 0$).
What is the $x$-intercept of the line $4x - 7y = -28$?
Solve the system of equations using substitution: $3x - y = 8$ and $x + 2y = 11$.
Using the elimination method, find the value of $y$ in the system: $5x + 3y = 2$ and $2x - 4y = -10$.
Determine the nature of the solutions for the system: $6x - 4y = 10$ and $3x - 2y = 8$.
Simplify the expression using positive exponents: $\left(\frac{3x^{-2}y^3}{x^4(y^{-1})^2}\right)^{-2}$.
Calculate the quotient: $\frac{6.3 \times 10^{-4}}{2.1 \times 10^5}$. Express the answer in scientific notation.
Simplify the expression: $(3a^2b^3)^2(2a^{-1}b^4)$.
Subtract $(3a^2 - 4ab + 5b^2)$ from $(7a^2 + 2ab - b^2)$.
Factor the polynomial completely: $4x^3 + 20x^2 - 24x$.
Factor the difference of squares completely: $x^4 - 81$.
Factor the trinomial completely: $6x^2 - 13x - 5$.
What is the vertex of the parabola defined by the function $y = -2(x + 5)^2 - 3$?
Solve the quadratic equation by factoring: $5x^2 - 20x = 60$.
Simplify the rational expression and state the simplified form: $\frac{2x^2 + 5x - 3}{4x^2 - 1}$.
Simplify the expression: $\frac{x^2 - 4}{x^2 + 5x + 6} \div \frac{x^2 - 4x + 4}{3x + 9}$.
Simplify the radical expression completely: $\sqrt{18x^5y^8} \cdot \sqrt{2x^3y^2}$, assuming $x \ge 0$ and $y \ge 0$.
Simplify the radical expression: $3\sqrt{50} - 5\sqrt{32} + \sqrt{2}$.
Rationalize the denominator of $\frac{4}{3 - \sqrt{5}}$. Simplify fully.
A factory produces Product A -4 hrs, 15dollars cost and Product B -2 hrs, 20 dollars cost. Total production time cannot exceed 2000 hours, and the budget is $12,000. Let A and B be the number of units of Product A and B, respectively. Assuming the factory is attempting to maximize output, which constraint is potentially the most binding [i.e., limits the production region most strictly]?
A motorboat travels 60 km upstream and immediately returns 60 km downstream. The total time taken for the round trip is 5 hours. If the speed of the river current is $5 \text{ km/h}$, determine the speed of the motorboat in still water.
A laboratory technician mixed Solution A, which is $40\%$ acid, with Solution B, which is $70\%$ acid, to produce 30 liters of a final solution that is $50\%$ acid. How many liters of Solution A ($40\%$ concentration) were used?
A certain two-digit number is five times the sum of its digits. If 9 is added to the number, the digits of the original number are reversed. What is the two-digit number?
In a survey of coffee shop customers, $60\%$ bought Lattes (L), $45\%$ bought Muffins (M), and $35\%$ bought Scones (S). We know $20\%$ bought L and M, $15\%$ bought M and S, $10\%$ bought L and S, and $5\%$ bought all three. What percentage of customers bought exactly one item?
In a corporation, the initial ratio of managers to non-managers is $2:13$. Over the next quarter, 8 new managers are hired and 2 non-managers leave, resulting in a new ratio of $1:5$. What was the original total number of employees in the corporation?
A rectangular garden measuring $10 \text{ m}$ by $15 \text{ m}$ is surrounded by a path of uniform width. If the total area of the path alone is $84 \text{ square meters}$, what is the width of the path?
A company sells gadgets for 50 dollars per unit. The monthly fixed overhead is 12,000 dollars, and the variable cost per unit is 30 dollars. What is the minimum number of units that must be sold monthly to achieve a profit of at least 6,000 dollars?
A delivery driver plans a round trip covering a total distance of $480 \text{ km}$. The car consumes fuel at a rate of 1 liter per $10 \text{ km}$. If the driver requires an additional $5 \text{ liters}$ safety reserve beyond the calculated need for the trip, what minimum fuel tank capacity (in liters) is required?
A complex rate problem involves two machines, $A$ and $B$, working together. Machine $A$ takes $1$ hour longer than Machine $B$ to complete a task independently. If $A$ and $B$ work together, they can complete the task in $1$ hour and $12$ minutes. If we let $t$ be the time, in hours, required for Machine $A$ to complete the task alone, which of the following quadratic equations correctly models this situation and what is the value of $t$?