Question 1

Solve the equation $$2(3y - 5) = 3(2y - 3) - 1$$ for $$y. $$This equation models a scenario where two competing companies' profits are equal after accounting for fixed and variable costs. What is the value of $$y$$?

Accepted Answer

1

Question 2

Solve the equation $$\frac{2}{3}(x - 1) = \frac{1}{2}(x + 2)$$ for $$x. $$Suppose $$x$$ represents the number of hours worked, and the equation models two payment plans. What is the value of $$x$$ where the plans pay equally?

Accepted Answer

4

Question 3

Solve the equation $$r = 2st - \frac{1}{3}ar$$ for $$a. $$This equation could represent a physics formula for resistance. What is $$a in $$terms of $$r$$, $$s$$, and $$t$$?

Accepted Answer

$$a = \frac{3r}{2st - r}$$

Question 4

Solve the equation $$|2x - 1| = 3$$ for $$x. $$Absolute value equations are used in engineering to measure tolerances. What are the possible values of $$x$$?

Accepted Answer

$$-1 or$$ $$2$$

Question 5

Solve the inequality $$|-2y + 3| + 2 \u003c 1$$ for $$y. $$This could represent a constraint in a manufacturing process. What is the solution?

Accepted Answer

No solution

Question 6

Solve the compound inequality $$2x + 3 \leq -1 or$$ $$-2x - 3 \leq -5$$ for $$x. $$Compound inequalities are used in budgeting scenarios. What is the solution set?

Accepted Answer

$$x \leq -2 or$$ $$x \geq 1$$

Question 7

Find the $$x-$$intercept of the graph of $$x - 3y = 6. $$The $$x-$$intercept is where a business's revenue equals zero. What is the $$x-$$intercept?

Accepted Answer

$$(6, 0)$$

Question 8

Find the equation of the line passing through the points $$(-1, -5)$$ and $$(2, 1). $$This models the change in temperature over time. What is the equation?

Accepted Answer

$$y = 2x -2$$

Question 9

Find the equation of the line that is perpendicular to $$x - 3y = 2$$ and passes through the point $$(2, 0). $$Perpendicular lines are used in navigation. What is the equation?

Accepted Answer

$$y = 3x -6$$

Question 10

Solve the following system of linear equations:

$$\begin{cases}
\frac{1}{9}x + \frac{1}{3}y = 1 \
\frac{1}{2}x - \frac{1}{4}y = 1
\end{cases}$$

Systems of equations are used in economics to model supply and demand. What is the solution $$(x, y)$$?

Accepted Answer

$$(x, y) = (4, -4)$$

Question 11

Expand $$(3y - 2)(y^2$ - 7y + 8). $$Polynomial expansion is used in calculating areas. What is the expanded form?

Accepted Answer

$$3y^3$$ - $$23y^2 + 38y - 16$

Question 12

Evaluate $$\sqrt{49} + \sqrt[3]{-27} - \sqrt[4]{16}. $$This calculation could represent a chemistry reaction's energy change. What is the result?

Accepted Answer

0

Question 13

Simplify $$(125a^{-6}b^9$)^{1/3}. $$This expression could represent a scaling law in biology. What is the simplified form?

Accepted Answer

$$\frac{5b^3$}{a^2$}$$

Question 14

Simplify $$16^{-\frac{3}$${4}}$$. Negative exponents are used in physics to model decay. What is the simplified value?

Accepted Answer

$$\frac{1}{32}$$

Question 15

Factor $$x^2 - 10x + 25$$ - $$16y^2. $$Factoring is used in engineering to optimize designs. What is the correct factorization?

Accepted Answer

$$(x + 4y - 5)(x - 4y + 5)$$

Question 16

If $$3x - 2 is a $$factor of $$6x^2 - 7x + 2$, what is the other factor? This is used in cryptography for polynomial division.

Accepted Answer

$$2x + 1$$

Question 17

For which of the following values is the rational expression $$\frac{x^2$ - 9}{x(x^2$ - x - 12)}$$ defined? Rational expressions are used in medicine for dosage calculations. Select the value where the expression is defined.

Accepted Answer

$$-4$$

Question 18

Which of the following equations can NOT be a function? Functions are used in computer science to model input-output relationships.

Accepted Answer

$$x = 3$$

Question 19

Find the domain of the function $$f(x) = \sqrt{3 - x}. $$Domains are important in biology for population models. What is the domain?

Accepted Answer

$$x \leq 3$$

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