Question 1

Given the inequality $$x^2 - 5x + 12$$ \geq 0$$, which of the following best describes the solution set?

Accepted Answer

All real numbers

Question 2

Solve the rational inequality $$\dfrac{x+4}{x-2} \geq 3$$ and select the correct solution set.

Accepted Answer

$$x \u003c 2 or$$ $$x \geq 7$$

Question 3

Which of the following is the correct factorization for $$a^3$$ - $$b^3$$?

Accepted Answer

$$(a-b)(a^2 + ab$$ + $$b^2$$)$$

Question 4

A chemical solution's concentration $$C (in $$mol/L) is modeled by $$C = \dfrac{t+4}{t-2}$$, where $$t is $$time in hours. For safety, the concentration must be at least 3 mol/L. For which values of $$t is $$the solution safe?

Accepted Answer

$$t \u003c 2 or$$ $$t \geq 7$$

Question 5

Which of the following is NOT a valid step when solving a rational inequality algebraically?

Accepted Answer

Multiplying both sides by the denominator without considering its sign

Question 6

Given the polynomial inequality $$x^2 - 5x + 12$$ \u003c 0$$, what is the solution set?

Accepted Answer

No real solution

Question 7

A company models its profit $$P (in $$thousands of dollars) by $$P = x^2$ - 5x + 12$$, where $$x is $$the number of products sold (in hundreds). For which values of $$x is $$the profit non-negative?

Accepted Answer

All real values of $$x$$

Question 8

When solving the inequality $$\dfrac{x+4}{x-2} \geq 3$$, which value must be excluded from the solution set?

Accepted Answer

$$x = 2$$

Question 9

Which of the following best describes the process for solving a rational inequality algebraically?

Accepted Answer

Write as a single fraction, find zeros and undefined points, test intervals, and select intervals based on the inequality sign

Question 10

If $$x$$ represents the number of pencils sold and the profit is modeled by $$P = x^2$ - 5x + 12$$, what is the minimum value of $$P$$?

Accepted Answer

$$P = 5.75 at$$ $$x = 2.5$$

Question 11

Which of the following is the correct factorization for $$a^3$$ + $$b^3$$?

Accepted Answer

$$(a+b)(a^2 - ab$$ + $$b^2$$)$$

Question 12

A rational function $$R(x) = \dfrac{x+4}{x-2}$$ models the rate of a chemical reaction. For which value of $$x$$ does the reaction rate become undefined?

Accepted Answer

$$x = 2$$

Question 13

Which of the following intervals is a solution to the inequality $$\dfrac{x+4}{x-2} \geq 3$$?

Accepted Answer

$$(-\infty, 2) \cup [7, \infty)$$

Question 14

If a rational inequality has a denominator that is zero at $$x = a$$, what should you do when writing the solution set?

Accepted Answer

Exclude $$x = a$$ from the solution set

Question 15

A business wants its profit, modeled by $$P = x^2$ - 5x + 12$$, to be at least $$20. $$For which values of $$x is $$this true?

Accepted Answer

No real values of $$x$$

Question 16

Which of the following is a critical step in solving polynomial inequalities?

Accepted Answer

Finding the zeros of the polynomial

Question 17

If $$x^2 - 5x + 12$ is always positive for all real x$, what does this tell you about its graph?

Accepted Answer

The parabola does not intersect the $$x-$$axis

Question 18

A rational function $$f(x) = \dfrac{x+4}{x-2} is $$used to model the speed of a vehicle. For which values of $$x is $$the speed at least 3 units?

Accepted Answer

$$x \u003c 2 or$$ $$x \geq 7$$

Solving Polynomial and Rational Inequalities (Precalculus 3.5 Study Notes)

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