Question 1

Which of the following is the correct first step when solving a polynomial or rational inequality algebraically?

Accepted Answer

Write the inequality so that a polynomial or rational function is on one side and zero is on the other side, in the form $$f(x) \geq 0$$, $$f(x) \leq 0$$, $$f(x) \u003e 0$$, or $$f(x) \u003c 0.$$

Question 2

Given the rational inequality $$\displaystyle \frac{x-2}{x+3} \u003e 0$$, which values must be excluded from the solution set?

Accepted Answer

$$x = -3$$, because the denominator is zero at this value.

Question 3

When solving the polynomial inequality $$x^3$$ - $$x^2 - 90x$$ \geq 0$$, what are the critical numbers that divide the real number line into intervals?

Accepted Answer

$$x = -9,\ 0,\ 10$$

Question 4

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

Accepted Answer

Find the zeros of the numerator and the values where the denominator is zero, then test intervals between these points.

Question 5

Suppose you are solving $$\displaystyle \frac{x-4}{x+2} \geq 0. $$Which intervals should you test for the sign of the function?

Accepted Answer

$$(-\infty, -2)$$, $$(-2, 4)$$, $$(4, \infty)$$

Question 6

A chemical reaction is modeled by the inequality $$x^2 - 5x + 6$$ \u003c 0$$, where x$ is the concentration of a reactant. For which values of x$ is the reaction rate negative?

Accepted Answer

$$2 \u003c x \u003c 3$$

Question 7

If $$f(x) = (x-2)(x+9)x$$, for which intervals is $$f(x) \leq 0$$?

Accepted Answer

$$-9 \leq x \leq 0$$ \ or \ $$0 \leq x \leq 2$$

Question 8

Which of the following is a necessary step when solving a rational inequality?

Accepted Answer

Identify values where the denominator is zero and exclude them from the solution.

Question 9

A company’s profit is modeled by $$P(x) = -2x^2$ + 8x + 10. $$For which values of $$x is $$the profit positive?

Accepted Answer

$$-1 \u003c x \u003c 5$$

Question 10

When testing intervals for the sign of a polynomial or rational function, what is the best strategy?

Accepted Answer

Choose any number within each interval and substitute it into the function.

Question 11

Which of the following is true about the solution to $$x^2 - 4$$ \leq 0$$?

Accepted Answer

$$-2 \leq x \leq 2$$

Question 12

A rational function $$f(x) = \frac{x+1}{x-5} is $$positive for which intervals?

Accepted Answer

$$x \u003e -1$$ and $$x \u003c 5$$

Question 13

Which of the following best describes the domain of the rational function $$f(x) = \frac{x-2}{x+3}$$?

Accepted Answer

All real numbers except $$x = -3$$

Question 14

If a rational inequality is written as $$\frac{p(x)}{q(x)} \leq 0$$, what must you do before testing intervals?

Accepted Answer

Find all real zeros of $$p(x)$$ and all real zeros of $$q(x).$$

Question 15

A medical dosage is safe if $$0.5x^2 - 3x + 2$$ \u003c 0$$. For which values of x$ is the dosage safe?

Accepted Answer

$$1 \u003c x \u003c 4$$

Question 16

Which of the following is NOT a correct step in solving a polynomial or rational inequality?

Accepted Answer

Multiplying both sides by the denominator without considering its sign.

Polynomial and Rational Inequalities: Methods and Examples

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