Question 1

Which of the following statements correctly describes the effect of multiplying both sides of an inequality by a negative constant?

Accepted Answer

The direction of the inequality is reversed.

Question 2

Given the inequality $$3 - 2x \u003e 7$$, what is the solution set?

Accepted Answer

$$x \u003c -2$$

Question 3

Suppose you are solving the inequality $$x^2 - x$$ \u003c 0$$. Which intervals contain the solution?

Accepted Answer

$$(0, 1)$$

Question 4

If $$f(x) = x^2$ - x$$, which of the following is true about the sign of $$f(x) on $$the interval $$(0, 1)$$?

Accepted Answer

$$f(x) \u003c 0$$

Question 5

Which of the following is the correct solution to the inequality $$x^2$$ \leq a$$, where a$$ \u003e 0$$?

Accepted Answer

$$-\sqrt{a} \leq x \leq \sqrt{a}$$

Question 6

A rational function $$f(x) = \frac{-8x - 3}{x + 1} is $$undefined at $$x = -1. $$What is the significance of this point when solving the inequality $$\frac{5}{x+1} \leq 8$$?

Accepted Answer

It is a boundary point that must be excluded from the solution set.

Question 7

Which of the following is the correct solution to $$\frac{5}{x+1} \leq 8$$?

Accepted Answer

$$(-\infty, -1) \cup (-1, -\frac{3}{8}]$$

Question 8

When using the sign diagram technique to solve $$x^2 - x$$ \u003c 0$$, which test points should you use to determine the sign of the function in each interval?

Accepted Answer

Any value in each interval between the zeros, such as $$x = -1$$, $$x = 0.5$$, and $$x = 2$$

Question 9

Which property is described by Theorem 1.8.11 regarding the sign of algebraic functions?

Accepted Answer

The sign of an algebraic function is constant on any interval containing no zeros or points of discontinuity.

Question 10

A medical researcher models the concentration of a drug in the bloodstream by $$C(t) = t^2$ - 4t + 3. $$For what values of $$t is $$the concentration below a critical threshold of $$2$$ units?

Accepted Answer

$$t  3$$

Question 11

Which of the following is the correct solution to the inequality $$x^2$$ \u003e a$$, where a$$ \u003e 0$$?

Accepted Answer

$$x  \sqrt{a}$$

Question 12

A chemical reaction rate is modeled by $$R(x) = x^2$ - 4. $$For which values of $$x is $$the reaction rate negative?

Accepted Answer

$$-2 \u003c x \u003c 2$$

Question 13

Which of the following is a correct application of the reciprocal rule for inequalities?

Accepted Answer

$$0 \u003c x \u003c y \iff 0 \u003c \frac{1}{y} \u003c \frac{1}{x}$$

Question 14

Suppose $$f(x) = x^4$ - 5x^2$ + 4. $$Using the sign diagram technique, for which intervals is $$f(x) \u003c 0$$?

Accepted Answer

$$( -2, -1 ) \cup (1, 2)$$

Question 15

A population model is given by $$P(t) = t^2$ - 6t + 8. $$For which values of $$t is $$the population below $$0$$?

Accepted Answer

$$t  4$$

Question 16

Which of the following is the correct solution to the inequality $$x^2 - 4x + 3$$ \geq 0$$?

Accepted Answer

$$x \leq 1 or$$ $$x \geq 3$$

Question 17

When solving $$x^2 - x$$ \u003e 0$$, which intervals are included in the solution?

Accepted Answer

$$(-\infty, 0) \cup (1, \infty)$$

Question 18

A rational function $$f(x) = \frac{2x+1}{x-3} is $$undefined at $$x = 3. $$When solving $$\frac{2x+1}{x-3} \u003e 0$$, which intervals should be considered?

Accepted Answer

$$(-\infty, -\frac{1}{2}) \cup (3, \infty)$$

Question 19

Which of the following is NOT a valid step in Procedure 1.8.12 for solving inequalities using sign diagrams?

Accepted Answer

Ignoring points where the function is undefined.

Elementary Inequality Rules and Sign Diagram Techniques in Calculus

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