Search in book...
Toggle Font Controls
Create new playlist

Name your new playlist

Playlist description (optional)
Sign In

Email address

Password

Forgot Password?

or

Continue with Facebook

Continue with Google
Sign Up

Full Name

Email address

Confirm Email Address

Password

or

Continue with Facebook

Continue with Google

CHAPTER 17 REVIEW EXERCISES

CONCEPT CHECK EXERCISES

Determine each of the following as being either true or false. If it is false, explain why.

$x < - 3$ $x < - 3$ or $x > 1$ $x > 1$ may also be written as $1 < x < - 3.$ $1 < x < - 3.$
The solution of the inequality $- 3 x > 6$ $- 3 x > 6$ is $x > - 2.$ $x > - 2.$
The solution of the inequality $x^{2} - 2 x - 8 > 0$ $x^{2} - 2 x - 8 > 0$ is $x < - 2$ $x < - 2$ or $x > 4.$ $x > 4.$
The solution of the inequality $| x - 2 | < 5$ $| x - 2 | < 5$ is $x < - 3$ $x < - 3$ or $x > 7.$ $x > 7.$
The graphical solution of the inequality $y \geq x + 1$ $y \geq x + 1$ is shown as all points above the line $y = x + 1.$ $y = x + 1.$
The maximum value of the objective function $F = x + 2 y,$ $F = x + 2 y,$ subject to the conditions $x \geq 0, y \geq 0, 2 x + 3 y \leq 6,$ $x \geq 0, y \geq 0, 2 x + 3 y \leq 6,$ is 4.

PRACTICE AND APPLICATIONS

In Exercises 7–22, solve each of the given inequalities algebraically. Graph each solution.

$2 x - 12 > 0$ $2 x - 12 > 0$
$2.4 (T - 4.0) \geq 5.5 - 2.4 T$ $2.4 (T - 4.0) \geq 5.5 - 2.4 T$
$4 < 2 x - 1 < 11$ $4 < 2 x - 1 < 11$
$2 x < x + 1 < 4 x + 7$ $2 x < x + 1 < 4 x + 7$
$5 x^{2} + 9 x < 2$ $5 x^{2} + 9 x < 2$
$x^{2} + 2 x > 63$ $x^{2} + 2 x > 63$
$6 n^{2} - n > 35$ $6 n^{2} - n > 35$
$2 x^{3} + 4 \leq x^{2} + 8 x$ $2 x^{3} + 4 \leq x^{2} + 8 x$
$\frac{(2 x - 1) (3 - x)}{x + 4} > 0$ $\frac{(2 x - 1) (3 - x)}{x + 4} > 0$
$x^{4} + x^{2} \leq 0$ $x^{4} + x^{2} \leq 0$
$\frac{8}{x} < 2$ $\frac{8}{x} < 2$
$\frac{1}{x - 2} > \frac{1}{4}$ $\frac{1}{x - 2} > \frac{1}{4}$
$| 3 x + 2 | \leq 4$ $| 3 x + 2 | \leq 4$
$| 4 - 3 x | \geq 7$ $| 4 - 3 x | \geq 7$
$| 3 - 5 x | > 7$ $| 3 - 5 x | > 7$
$3 | 2 - \frac{y}{2} | \leq 0$ $3 | 2 - \frac{y}{2} | \leq 0$

In Exercises 23–30, solve the given inequalities on a calculator such that the display is the graph of the solution.

$5 - 3 x > 0$ $5 - 3 x > 0$
$6 \leq 2 - 4 x < 9$ $6 \leq 2 - 4 x < 9$
$2 \leq \frac{4 n - 2}{3} < 3$ $2 \leq \frac{4 n - 2}{3} < 3$
$3 x < 2 x + 1 < x - 5$ $3 x < 2 x + 1 < x - 5$
$\frac{8 - R}{2 R + 1} \leq 0$ $\frac{8 - R}{2 R + 1} \leq 0$
$\frac{{(3 - x)}^{2}}{2 x + 7} \leq 0$ $\frac{{(3 - x)}^{2}}{2 x + 7} \leq 0$
$| x - 30 | > 48$ $| x - 30 | > 48$
$2 | 2 x - 9 | < 8$ $2 | 2 x - 9 | < 8$

In Exercises 31–34, use a calculator to solve the given inequalities. Graph the appropriate function and from the graph determine the solution.

$x^{3} + x + 1 < 0$ $x^{3} + x + 1 < 0$
$\frac{4}{R + 2} > 6$ $\frac{4}{R + 2} > 6$
$e^{- t} > 0.5$ $e^{- t} > 0.5$
$\sin 2 x < 0.8 (0 < x < 4)$ $\sin 2 x < 0.8 (0 < x < 4)$

In Exercises 35–46, draw a sketch of the region in which the points satisfy the given inequality or system of inequalities.

$y > 12 - 3 x$ $y > 12 - 3 x$
$y < \frac{1}{2} x + 2$ $y < \frac{1}{2} x + 2$
$4 y - 6 x - 8 \leq 0$ $4 y - 6 x - 8 \leq 0$
$3 y - x + 6 \geq 0$ $3 y - x + 6 \geq 0$
$2 x > 6 - y$ $2 x > 6 - y$
$y \leq \frac{6}{x^{2} - 49}$ $y \leq \frac{6}{x^{2} - 49}$
$y - | x + 1 | < 0$ $y - | x + 1 | < 0$
$2 y + 2 x^{3} + 6 x > 3$ $2 y + 2 x^{3} + 6 x > 3$
$\begin{array}{l} y > x + 1 \\ y < 4 - x^{2} \end{array}$ $\begin{array}{l} y > x + 1 \\ y < 4 - x^{2} \end{array}$
$\begin{array}{l} y > 2 x - x^{2} \\ y \geq - 2 \end{array}$ $\begin{array}{l} y > 2 x - x^{2} \\ y \geq - 2 \end{array}$
$\begin{array}{l} y \leq \frac{4}{x^{2} + 1} \\ y < x - 3 \end{array}$ $\begin{array}{l} y \leq \frac{4}{x^{2} + 1} \\ y < x - 3 \end{array}$
$\begin{array}{l} y < \cos \frac{1}{2} x \\ y > \frac{1}{2} e^{x} \\ - π < x < π \end{array}$ $\begin{array}{l} y < \cos \frac{1}{2} x \\ y > \frac{1}{2} e^{x} \\ - π < x < π \end{array}$

In Exercises 47–54, use a calculator to display the region in which the points satisfy the given inequality or system of inequalities.

$y < 3 x + 5$ $y < 3 x + 5$
$x > 8 - 4 y$ $x > 8 - 4 y$
$y > 8 + 7 x - x^{2}$ $y > 8 + 7 x - x^{2}$
$y < x^{3} + 4 x^{2} - x - 4$ $y < x^{3} + 4 x^{2} - x - 4$
$y < 32 x - x^{4}$ $y < 32 x - x^{4}$
$\begin{array}{l} y > 2 x - 1 \\ y < 6 - 3 x^{2} \end{array}$ $\begin{array}{l} y > 2 x - 1 \\ y < 6 - 3 x^{2} \end{array}$
$\begin{array}{l} y > 1 - x \sin 2 x \\ y < 5 - x^{2} \end{array}$ $\begin{array}{l} y > 1 - x \sin 2 x \\ y < 5 - x^{2} \end{array}$
$\begin{array}{l} y > | x - 1 | \\ y < 4 + \ln x \end{array}$ $\begin{array}{l} y > | x - 1 | \\ y < 4 + \ln x \end{array}$

In Exercises 55–58, determine the values of x for which the given radicals represent real numbers.

$\sqrt{4 - x}$ $\sqrt{4 - x}$
$\sqrt{x + 5}$ $\sqrt{x + 5}$
$\sqrt{x^{2} + 3 x}$ $\sqrt{x^{2} + 3 x}$
$\sqrt{\frac{x - 1}{2 x + 5}}$ $\sqrt{\frac{x - 1}{2 x + 5}}$

In Exercises 59–62, find the indicated maximum and minimum values by the method of linear programming. The constraints are shown below the objective function.

Maximum P:

$\begin{array}{l} P = 2 x + 9 y \\ x \geq 0, y \geq 0 \\ x + 4 y \leq 13 \\ 3 y - x \leq 8 \end{array}$ $\begin{array}{l} P = 2 x + 9 y \\ x \geq 0, y \geq 0 \\ x + 4 y \leq 13 \\ 3 y - x \leq 8 \end{array}$
Maximum P:

$\begin{array}{l} P = x + 2 y \\ x \geq 1, y \geq 0 \\ 3 x + y \leq 6 \\ 2 x + 3 y \leq 8 \end{array}$ $\begin{array}{l} P = x + 2 y \\ x \geq 1, y \geq 0 \\ 3 x + y \leq 6 \\ 2 x + 3 y \leq 8 \end{array}$
Minimum C:

$\begin{array}{l} C = 3 x + 4 y \\ x \geq 0, y \geq 1 \\ 2 x + 3 y \geq 6 \\ 4 x + 2 y \geq 5 \end{array}$ $\begin{array}{l} C = 3 x + 4 y \\ x \geq 0, y \geq 1 \\ 2 x + 3 y \geq 6 \\ 4 x + 2 y \geq 5 \end{array}$
Minimum C:

$\begin{array}{l} C = 2 x + 4 y \\ x \geq 0, y \geq 0 \\ x + 3 y \geq 6 \\ 4 x + 7 y \geq 18 \end{array}$ $\begin{array}{l} C = 2 x + 4 y \\ x \geq 0, y \geq 0 \\ x + 3 y \geq 6 \\ 4 x + 7 y \geq 18 \end{array}$

In Exercises 63–90, solve the given problems using inequalities. (All data are accurate to at least two significant digits.)

Under what conditions is $| a + b | < | a | + | b | ?$ $| a + b | < | a | + | b | ?$
Is $| a - b | < | a | + | b |$ $| a - b | < | a | + | b |$ always true? Explain.
For what values of x is the graph of $y = x^{2} + 3 x$ $y = x^{2} + 3 x$ above the graph of $y = 2 x + 6 ?$ $y = 2 x + 6 ?$
Solve for x: $a + | b x | < c$ $a + | b x | < c$ given that $a - c < 0.$ $a - c < 0.$
Form an inequality of the form $a x^{2} + b x + c < 0$ $a x^{2} + b x + c < 0$ with $a > 0$ $a > 0$ for which the solution is $- 3 < x < 5.$ $- 3 < x < 5.$
By means of an inequality, define the region above the line $x - 3 y - 6 = 0.$ $x - 3 y - 6 = 0.$
Draw a graph of the system $y < 1 - x^{2}$ $y < 1 - x^{2}$ and $y = x^{2} .$ $y = x^{2} .$
Find a system of inequalities that would describe the region within the quadrilateral with vertices (0, 0), (4, 4), $(0, - 3),$ $(0, - 3),$ and $(4, - 3) .$ $(4, - 3) .$
Find the values for which $f (x) = (x - 2) (x - 3)$ $f (x) = (x - 2) (x - 3)$ is positive, zero, and negative. Use this information along with f(0) and f(5) to make a rough sketch of the graph of f(x).
Follow the same instructions as in Exercise 71 for the function $f (x) = (x - 2) / (x - 3) .$ $f (x) = (x - 2) / (x - 3) .$
Describe the region that satisfies the system $y \geq 2 x - 3, y < 2 x - 3.$ $y \geq 2 x - 3, y < 2 x - 3.$
Describe the region defined by $y \geq 2 x - 3 or y < 2 x - 3.$ $y \geq 2 x - 3 or y < 2 x - 3.$
If two adjacent sides of a square design on a TV screen expand 6.0 cm and 10.0 cm, respectively, how long is each side of the original square if the perimeter of the resulting rectangle is at least twice that of the original square? See Fig. 17.48.

Fig. 17.48
The value V (in $) of each building lot in a development is estimated as $V = 65, 000 + 5000 t,$ $V = 65, 000 + 5000 t,$ where t is the time in years from now. For how long is the value of each lot no more than $90,000?
The cost C of producing two of one type of calculator and five of a second type is $50. If the cost of producing each of the second type is between $5 and $8, what are the possible costs of producing each of the first type?
City A is 600 km from city B. One car starts from A for B 1 h before a second car. The first car averages 60 km/h, and the second car averages 80 km/h for the trip. For what times after the first car starts is the second car ahead of the first car?
The pressure p (in kPa) at a depth d (in m) in the ocean is given by $p = 101 + 10.1 d .$ $p = 101 + 10.1 d .$ For what values of d is $p > 500 kPa ?$ $p > 500 kPa ?$
After conducting tests, it was determined that the stopping distance x (in ft) of a car traveling 60 mi/h was $| x - 290 | \leq 35.$ $| x - 290 | \leq 35.$ Express this inequality without absolute values and find the interval of stopping distances that were found in the tests.
A heating unit with 80% efficiency and a second unit with 90% efficiency deliver 360,000 Btu of heat to an office complex. If the first unit consumes an amount of fuel that contains no more than 261,000 Btu, what is the Btu content of the fuel consumed by the second unit?
A rectangular parking lot is to have a perimeter of 180 m and an area of at least $2000 m^{2} .$ $2000 m^{2} .$ What are the possible dimensions of the lot?
The electric power p (in W) dissipated in a resistor is given by $p = R i^{2},$ $p = R i^{2},$ where R is the resistance $(in Ω)$ $(in Ω)$ and i is the current (in A). For a given resistor, $R = 12.0 Ω,$ $R = 12.0 Ω,$ and the power varies between 2.50 W and 8.00 W. Find the values of the current.
The reciprocal of the total resistance of two electric resistances in parallel equals the sum of the reciprocals of the resistances. If a $2.0 - Ω$ $2.0 - Ω$ resistance is in parallel with a resistance R, with a total resistance greater than $0.5 Ω,$ $0.5 Ω,$ find R.
The efficiency e (in %) of a certain gasoline engine is given by $e = 100 (1 - r^{- 0.4}),$ $e = 100 (1 - r^{- 0.4}),$ where r is the compression ratio for the engine. For what values of r is $e > 50 % ?$ $e > 50 % ?$
A rocket is fired such that its height h (in mi) is given by $h = 41 t - t^{2} .$ $h = 41 t - t^{2} .$ For what values of t (in min) is the height greater than 400 mi?
In developing a new product, a company estimates that it will take no more than 1200 min of computer time for research and no more than 1000 min of computer time for development. Graph the possible combinations of the computer times that are needed.
A natural gas supplier has a maximum of 120 worker-hours per week for delivery and for customer service. Graph the possible combinations of times available for these two services.
A company produces two types of cell phones, the regular model and the deluxe model. For each regular model produced, there is a profit of $8, and for each deluxe model the profit is $15. The same amount of materials is used to make each model, but the supply is sufficient only for 450 cell phones per day. The deluxe model requires twice the time to produce as the regular model. If only regular models were made, there would be time enough to produce 600 per day. Assuming all cell phones will be sold, how many of each model should be produced if the profit is to be a maximum?
A company that manufactures DVD/CD players gets two different parts, A and B, from two different suppliers. Each package of parts from the first supplier costs $2.00 and contains 6 of each type of part. Each package of parts from the second supplier costs $1.50 and contains 4 of A and 8 of B. How many packages should be bought from each supplier to keep the total cost to a minimum, if production requirements are 600 of A and 900 of B?
In planning a new city development, an engineer uses a rectangular coordinate system to locate points within the development. A park in the shape of a quadrilateral has corners at (0, 0), (0, 20), (40, 20), and (20, 40) (measurements in meters). Write two or three paragraphs explaining how to describe the park region with inequalities and find these inequalities.

..................Content has been hidden....................

You can't read the all page of ebook, please click here login for view all page.

Table of Contents for CHAPTER 17 REVIEW EXERCISES

Create new playlist

Sign In

Sign Up

Table of Contents for
CHAPTER 17 REVIEW EXERCISES