Question 1

Which of the following is the correct first step in solving the polynomial inequality $$x^3 - 4x$$ \u003c 0$$?

Accepted Answer

Factor the polynomial and set it equal to zero.

Question 2

Given the polynomial inequality $$x^2 - 5x + 6$$ \u003e 0$$, what are the boundary points?

Accepted Answer

$$x = 2$$ and $$x = 3$$

Question 3

After finding the boundary points for a polynomial inequality, what is the next step?

Accepted Answer

Test values in each interval to determine the sign of the polynomial.

Question 4

Solve the rational inequality $$\frac{x-2}{x+3} \u003e 0. $$Which intervals are part of the solution set?

Accepted Answer

$$(-\infty, -3) \cup (2, \infty)$$

Question 5

Which of the following best describes a rational inequality?

Accepted Answer

An inequality that can be written in the form $$\frac{f(x)}{g(x)} \u003c 0$$ where $$f$$ and $$g$$ are polynomials.

Question 6

A ball is thrown upward from ground level with an initial velocity of 64 ft/s. Its height after $$t$$ seconds is given by $$s(t) = -16t^2$ + 64t. $$For which values of $$t is $$the ball at least 48 feet above the ground?

Accepted Answer

$$1 \u003c t \u003c 3$$

Question 7

When solving the inequality $$x^2 - 4$$ \leq 0$$, which of the following is the correct solution set?

Accepted Answer

$$[-2, 2]$$

Question 8

Which of the following is a necessary step when solving a rational inequality such as $$\frac{x+1}{x-2} \leq 0$$?

Accepted Answer

Find the zeros of both the numerator and denominator.

Question 9

A chemical reaction requires the temperature $$T (in $$degrees Celsius) to satisfy $$T^2 - 9T + 20$$ \u003c 0$$ for optimal results. What is the range of acceptable temperatures?

Accepted Answer

$$4 \u003c T \u003c 5$$

Question 10

Which of the following intervals is the solution to the inequality $$x^3$$ - $$x^2 - 6x$$ \u003e 0$$?

Accepted Answer

$$(-\infty, -2) \cup (0, 3)$$

Question 11

A medication is effective only if the concentration $$C (in mg/L) $$satisfies $$\frac{C-2}{C+1} \u003c 0. $$What is the range of effective concentrations?

Accepted Answer

$$-1 \u003c C \u003c 2$$

Question 12

Which of the following best describes the role of test values when solving polynomial or rational inequalities?

Accepted Answer

They determine the sign of the function in each interval between boundary points.

Question 13

Solve the inequality $$x^2 + 2x - 8$$ \geq 0$$. What is the solution set?

Accepted Answer

$$(-\infty, -4] \cup [2, \infty)$$

Question 14

A projectile is launched from a height of 32 feet with an initial velocity of 48 ft/s. Its height after $$t$$ seconds is $$s(t) = -16t^2$ + 48t + 32. $$For which values of $$t is $$the projectile at least 64 feet above the ground?

Accepted Answer

$$1 \u003c t \u003c 3$$

Question 15

Which of the following is NOT a valid step in solving a polynomial inequality?

Accepted Answer

Multiplying both sides by a variable expression without considering its sign.

Polynomial and Rational Inequalities: Study Notes for College Algebra

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