Chapter 17

1. $x < 0, y > 0$ $x < 0, y > 0$
2.
3. $\begin{array}{rcl} 3 x + 1 & < & - 5 \\ 3 x & < & - 6 \\ x & < & - 2 \end{array}$ $\begin{array}{rcl} 3 x + 1 & < & - 5 \\ 3 x & < & - 6 \\ x & < & - 2 \end{array}$
4. $\begin{array}{rcl} - 1 & < & 1 - 2 x < 5 \\ - 2 & < & - 2 x < 4 \\ 1 & > & x > - 2 \\ - 2 & < & x < 1 \end{array}$ $\begin{array}{rcl} - 1 & < & 1 - 2 x < 5 \\ - 2 & < & - 2 x < 4 \\ 1 & > & x > - 2 \\ - 2 & < & x < 1 \end{array}$

5. $\frac{x^{2} + x}{x - 2} \leq 0, \frac{x (x + 1)}{x - 2} \leq 0$ $\frac{x^{2} + x}{x - 2} \leq 0, \frac{x (x + 1)}{x - 2} \leq 0$

Interval	$\frac{x (x + 1)}{x - 2}$ $\frac{x (x + 1)}{x - 2}$	Sign
$x < - 1$ $x < - 1$	$\frac{- -}{-}$ $\frac{- -}{-}$	$-$ $-$
$- 1 < x < 0$ $- 1 < x < 0$	$\frac{- +}{-}$ $\frac{- +}{-}$	$+$ $+$
$0 < x < 2$ $0 < x < 2$	$\frac{+ +}{-}$ $\frac{+ +}{-}$	$-$ $-$
$x > 2$ $x > 2$	$\frac{+ +}{+}$ $\frac{+ +}{+}$	$+$ $+$

Solution: $x \leq - 1$ $x \leq - 1$ or $0 \leq x < 2$ $0 \leq x < 2$

A number line shaded to the left of a closed circle at negative 1, and shaded between a closed circle at 0 and open circle at 2.

(x cannot equal 2)

6. $\begin{array}{rclrcl} | 2 x + 1 | & \geq & 3 \\ 2 x + 1 & \geq & 3 or & 2 x + 1 & \leq & - 3 \\ 2 x & \geq & 2 or & 2 x & \leq & - 4 \\ x & \geq & 1 or & x & \leq & - 2 \end{array}$ $\begin{array}{rclrcl} | 2 x + 1 | & \geq & 3 \\ 2 x + 1 & \geq & 3 or & 2 x + 1 & \leq & - 3 \\ 2 x & \geq & 2 or & 2 x & \leq & - 4 \\ x & \geq & 1 or & x & \leq & - 2 \end{array}$
7. $\begin{array}{l} | 2 - 3 x | < 8 \\ - 8 < 2 - 3 x < 8 \\ - 10 < - 3 x < 6 \\ \frac{10}{3} > x > - 2 \\ - 2 < x < \frac{10}{3} \end{array}$ $\begin{array}{l} | 2 - 3 x | < 8 \\ - 8 < 2 - 3 x < 8 \\ - 10 < - 3 x < 6 \\ \frac{10}{3} > x > - 2 \\ - 2 < x < \frac{10}{3} \end{array}$
8.

9. If $\sqrt{x^{2} - x - 6}$ $\sqrt{x^{2} - x - 6}$ is real, then $x^{2} - x - 6 \geq 0.$ $x^{2} - x - 6 \geq 0.$

$(x - 3) (x + 2) \geq 0$ $(x - 3) (x + 2) \geq 0$

$x \leq - 2$ $x \leq - 2$ or $x \geq 3$ $x \geq 3$

Interval	$(x - 3) (x + 2) \frac{}{}$ $(x - 3) (x + 2) \frac{}{}$	Sign
$x < - 2$ $x < - 2$	$(-) (-) \frac{}{}$ $(-) (-) \frac{}{}$	$+$ $+$
$- 2 < x < 3$ $- 2 < x < 3$	$(-) (+) \frac{}{}$ $(-) (+) \frac{}{}$	$-$ $-$
$x > 3$ $x > 3$	$(+) (+) \frac{}{}$ $(+) (+) \frac{}{}$	$+$ $+$

10. Let $w = width, l = length$ $w = width, l = length$

$\begin{array}{rclrcl} l & = & w + 20 & w^{2} + 20 w - 4800 & \geq & 0 \\ w l & \geq & 4800 & (w + 80) (w - 60) & \geq & 0 \\ w (w + 20) & \geq & 4800 & w & \geq & 60 m \end{array}$ $\begin{array}{rclrcl} l & = & w + 20 & w^{2} + 20 w - 4800 & \geq & 0 \\ w l & \geq & 4800 & (w + 80) (w - 60) & \geq & 0 \\ w (w + 20) & \geq & 4800 & w & \geq & 60 m \end{array}$
11. $\begin{array}{l} \begin{array}{rcl} Let A & = & length of type A wire \\ B & = & length of type B wire \end{array} \\ 0.10 A + 0.20 B < 5.00 \\ A + 2 B < 50 \end{array}$ $\begin{array}{l} \begin{array}{rcl} Let A & = & length of type A wire \\ B & = & length of type B wire \end{array} \\ 0.10 A + 0.20 B < 5.00 \\ A + 2 B < 50 \end{array}$
12. $| λ - 550 nm | < 150 nm (within 150 nm of λ = 550 nm)$ $| λ - 550 nm | < 150 nm (within 150 nm of λ = 550 nm)$
13. $x^{2} > 12 - x$ $x^{2} > 12 - x$

$x < - 4, x > 3$ $x < - 4, x > 3$

14. $\begin{array}{l} P = 5 x + 3 y \\ x \geq 0, y \geq 0 \\ 2 x + 3 y \leq 12 \\ 4 x + y \leq 8 \end{array}$ $\begin{array}{l} P = 5 x + 3 y \\ x \geq 0, y \geq 0 \\ 2 x + 3 y \leq 12 \\ 4 x + y \leq 8 \end{array}$

A solid line falls through (six-fifths, sixteen-fifths) to (2, 0), and another solid line falls from (0, 4) through (six-fifths, sixteen-fifths). The area below both lines and within quadrant 1 is shaded.

Vertices:	(0,0)	(2,0)	$(\frac{6}{5}, \frac{16}{5})$ $(\frac{6}{5}, \frac{16}{5})$	(0,4)
$P = 5 x + 3 y$ $P = 5 x + 3 y$	0	10	15.6	12

Max. value of $P = 15.6$ $P = 15.6$ at $(\frac{6}{5}, \frac{16}{5})$ $(\frac{6}{5}, \frac{16}{5})$

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Table of Contents for Chapter 17

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Table of Contents for
Chapter 17