Question 1

Which of the following is the correct first step in solving the polynomial inequality $$2x^2 + x$$ \u003e 15$$?

Accepted Answer

Rewrite as $$2x^2 + x - 15$$ \u003e 0$$

Question 2

After expressing $$2x^2 + x - 15$$ \u003e 0$$, what is the next step in solving the inequality?

Accepted Answer

Solve $$2x^2 + x - 15 = 0$ for x$

Question 3

What are the boundary points for the inequality $$2x^2 + x - 15$$ \u003e 0$$?

Accepted Answer

$$x = -3$$ and $$x = \frac{5}{2}$$

Question 4

Given the intervals $$(-\infty, -3)$$, $$(-3, \frac{5}{2})$$, and $$(\frac{5}{2}, \infty)$$, which interval(s) satisfy $$2x^2 + x - 15$$ \u003e 0$$?

Accepted Answer

$$(-\infty, -3) \cup (\frac{5}{2}, \infty)$$

Question 5

When solving a rational inequality, why must you set both the numerator and denominator equal to zero to find boundary points?

Accepted Answer

Because zeros of the numerator make the expression zero and zeros of the denominator make it undefined

Question 6

Solve the rational inequality $$\frac{x-5}{x+3} \geq 0. $$What is the solution set?

Accepted Answer

$$(-5, -3) \cup [5, \infty)$$

Question 7

Which of the following best describes the process of testing intervals when solving polynomial or rational inequalities?

Accepted Answer

Choose a test value from each interval and substitute it into the original function to determine the sign

Question 8

A ball is thrown upward and its height $$h in $$feet after $$t$$ seconds is given by $$h(t) = -16t^2$ + 48t + 32. $$For what values of $$t is $$the ball more than 64 feet above the ground?

Accepted Answer

$$t \in (1, 4)$$

Question 9

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

Question 10

If you solve $$f(x) \leq 0$$ and one of the boundary points makes the denominator zero, what should you do?

Accepted Answer

Exclude that boundary point from the solution set

Question 11

Given the inequality $$x^2 - 4x + 3$$ \u003c 0$$, what is the solution set?

Accepted Answer

$$(1, 3)$$

Question 12

Which of the following real-world scenarios can be modeled by a polynomial or rational inequality?

Accepted Answer

Determining when a car's speed exceeds a legal limit based on a quadratic speed function

Question 13

If $$f(x) = (x-2)(x+1)$$, for which intervals is $$f(x) \u003c 0$$?

Accepted Answer

$$(-1, 2)$$

Question 14

When solving $$\frac{x+2}{x-1} \u003e 0$$, which of the following is true about the solution set?

Accepted Answer

$$(-2, 1)$$ and $$(1, \infty)$$, excluding $$x=1$$

Question 15

Which of the following best explains why boundary points where the denominator is zero are excluded from the solution set of a rational inequality?

Accepted Answer

Because the function is undefined at those points

Polynomial and Rational Inequalities: Methods and Applications

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