Question 1

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

Accepted Answer

Subtract 15 from both sides to obtain $$2x^2 + x - 15$$ \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$ to find the boundary points.

Question 3

What are the solutions to $$2x^2 + x - 15 = 0$?

Accepted Answer

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

Question 4

When solving $$2x^2 + x - 15$$ \u003e 0$$, which intervals should be tested for sign analysis?

Accepted Answer

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

Question 5

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

Accepted Answer

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

Question 6

Which of the following best describes the solution set for $$2x^2 + x$$ \u003e 15$$?

Accepted Answer

$$x  \frac{5}{2}$$

Question 7

When solving the rational inequality $$\frac{x+1}{x+3} \geq 2$$, what is the first algebraic step?

Accepted Answer

Subtract 2 from both sides to obtain $$\frac{x+1}{x+3} - 2 \geq 0.$$

Question 8

After simplifying $$\frac{x+1}{x+3} - 2 \geq 0$$, what is the resulting inequality?

Accepted Answer

$$\frac{-x-5}{x+3} \geq 0$$

Question 9

What are the boundary points for the rational inequality $$\frac{-x-5}{x+3} \geq 0$$?

Accepted Answer

$$x = -5$$ and $$x = -3$$

Question 10

Which intervals should be tested for sign analysis when solving $$\frac{-x-5}{x+3} \geq 0$$?

Accepted Answer

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

Question 11

For the rational inequality $$\frac{-x-5}{x+3} \geq 0$$, which interval(s) satisfy the inequality?

Accepted Answer

$$(-5, -3)$$

Question 12

Which boundary point(s) are included in the solution set for $$\frac{-x-5}{x+3} \geq 0$$?

Accepted Answer

$$x = -5$$ only

Question 13

A ball is thrown upward and its height in feet after $$t$$ seconds is given by $$s(t) = -16t^2$ + 80t + 48. $$For what values of $$t is $$the ball more than 64 feet above the ground?

Accepted Answer

$$1 \u003c t \u003c 4$$

Question 14

Which of the following best describes the general procedure for solving a polynomial inequality?

Accepted Answer

Express the inequality in standard form, solve for zeros, test intervals, and write the solution set.

Question 15

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

Accepted Answer

To find all boundary points where the expression is zero or undefined.

Question 16

Which of the following is a common mistake when solving polynomial or rational inequalities?

Accepted Answer

Failing to test intervals between boundary points.

Polynomial and Rational Inequalities: Methods and Applications

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