14.2.2: Chapter 2

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2.1 The Rectangular Coordinate Systems and Graphs

1.

$x$	$y=\frac{1}{2}x+2$	$\left(x,y\right)$
$\mathrm{-2}$	$y=\frac{1}{2}\left(\mathrm{-2}\right)+2=1$	$\left(\mathrm{-2},1\right)$
$\mathrm{-1}$	$y=\frac{1}{2}\left(\mathrm{-1}\right)+2=\frac{3}{2}$	$\left(-1,\frac{3}{2}\right)$
$0$	$y=\frac{1}{2}\left(0\right)+2=2$	$\left(0,2\right)$
$1$	$y=\frac{1}{2}\left(1\right)+2=\frac{5}{2}$	$\left(1,\frac{5}{2}\right)$
$2$	$y=\frac{1}{2}\left(2\right)+2=3$	$\left(2,3\right)$

2.

x-intercept is $\left(4,0\right);$ y-intercept is $\left(0,3\right).$

3.

$\sqrt{125}=5\sqrt{5}$

4.

$\left(-5,\frac{5}{2}\right)$

2.2 Linear Equations in One Variable

1.

$x=\mathrm{-5}$

2.

$x=\mathrm{-3}$

3.

$x=\frac{10}{3}$

4.

$x=1$

5.

$x=-\frac{7}{17}.$ Excluded values are $x=-\frac{1}{2}$ and $x=-\frac{1}{3}.$

6.

$x=\frac{1}{3}$

7.

$m=-\frac{2}{3}$

8.

$y=4x\mathrm{-3}$

9.

$x+3y=2$

10.

Horizontal line: $y=2$

11.

Parallel lines: equations are written in slope-intercept form.

12.

$y=5x+3$

2.3 Models and Applications

1.

11 and 25

2.

$C=2.5x+3,650$

3.

45 mi/h

4.

$L=37$ cm, $W=18$ cm

5.

250 ft²

2.4 Complex Numbers

1.

$\sqrt{\mathrm{-24}}=0+2i\sqrt{6}$

2.

3.

$(3\mathrm{-4}i)-(2+5i)=1\mathrm{-9}i$

4.

$\frac{5}{2}-i$

5.

$18+i$

6.

$\mathrm{-3}\mathrm{-4}i$

7.

$\mathrm{-1}$

2.5 Quadratic Equations

1.

$\left(x-6\right)\left(x+1\right)=0;x=6,x=-1$

2.

$\left(x\mathrm{-7}\right)\left(x+3\right)=0,$ $x=7,$ $x=\mathrm{-3.}$

3.

$\left(x+5\right)\left(x\mathrm{-5}\right)=0,$ $x=\mathrm{-5},$ $x=5.$

4.

$\left(3x+2\right)\left(4x+1\right)=0,$ $x=-\frac{2}{3},$ $x=-\frac{1}{4}$

5.

$x=0,x=\mathrm{-10},x=\mathrm{-1}$

6.

$x=4\pm \sqrt{5}$

7.

$x=3\pm \sqrt{22}$

8.

$x=-\frac{2}{3},$ $x=\frac{1}{3}$

9.

$5$ units

2.6 Other Types of Equations

1.

$\frac{1}{4}$

2.

$25$

3.

$\left\{\mathrm{-1}\right\}$

4.

$0,$ $\frac{1}{2},$ $-\frac{1}{2}$

5.

$1;$ extraneous solution $-\frac{2}{9}$

6.

$\mathrm{-2};$ extraneous solution $\mathrm{-1}$

7.

$\mathrm{-1},$ $\frac{3}{2}$

8.

$\mathrm{-3},3,-i,i$

9.

$2,12$

10.

$\mathrm{-1},$ $0$ is not a solution.

2.7 Linear Inequalities and Absolute Value Inequalities

1.

$\left[\mathrm{-3},5\right]$

2.

$\left(-\infty ,\mathrm{-2}\right)\cup \left[3,\infty \right)$

3.

$x<1$

4.

$x\ge \mathrm{-5}$

5.

$\left(2,\infty \right)$

6.

$\left[-\frac{3}{14},\infty \right)$

7.

$6<x\le 9\text{}\text{or}\left(6,9\right]$

8.

$\left(-\frac{1}{8},\frac{1}{2}\right)$

9.

$\left|x\mathrm{-2}\right|\le 3$

10.

$k\le 1$ or $k\ge 7;$ in interval notation, this would be $(-\infty ,1]\cup [7,\infty ).$

2.1 Section Exercises

1.

Answers may vary. Yes. It is possible for a point to be on the x-axis or on the y-axis and therefore is considered to NOT be in one of the quadrants.

3.

The y-intercept is the point where the graph crosses the y-axis.

5.

The x-intercept is $\left(2,0\right)$ and the y-intercept is $\left(0,6\right).$

7.

The x-intercept is $\left(2,0\right)$ and the y-intercept is $\left(0,\mathrm{-3}\right).$

9.

The x-intercept is $\left(3,0\right)$ and the y-intercept is $\left(0,\frac{9}{8}\right).$

11.

$y=4-2x$

13.

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

15.

$y=2x-\frac{4}{5}$

17.

$d=\sqrt{74}$

19.

$d=\sqrt{36}=6$

21.

$d\approx 62.97$

23.

$\left(3,\frac{-3}{2}\right)$

25.

$\left(2,\mathrm{-1}\right)$

27.

$\left(0,0\right)$

29.

$y=0$

31.

not collinear

33.

$\text{A:}\left(\mathrm{-3},2\right),\text{B:}\left(1,3\right),\text{C:}\left(4,0\right)$

35.

$x$	$y$
$\mathrm{-3}$	1
0	2
3	3
6	4

37.

x	y
–3	0
0	1.5
3	3

39.

41.

43.

$d=8.246$

45.

$d=5$

47.

$\left(\mathrm{-3},4\right)$

49.

$x=0y=\mathrm{-2}$

51.

$x=0.75y=0$

53.

$x=-1.667y=0$

55.

$15-11.2=3.8\text{mi}$ shorter

57.

$\text{6}.0\text{42}$

59.

Midpoint of each diagonal is the same point $(2,\mathrm{–2})$. Note this is a characteristic of rectangles, but not other quadrilaterals.

61.

37mi

63.

54 ft

2.2 Section Exercises

1.

It means they have the same slope.

3.

The exponent of the $x$ variable is 1. It is called a first-degree equation.

5.

If we insert either value into the equation, they make an expression in the equation undefined (zero in the denominator).

7.

$x=2$

9.

$x=\frac{2}{7}$

11.

$x=6$

13.

$x=3$

15.

$x=\mathrm{-14}$

17.

$x\ne \mathrm{-4};$ $x=\mathrm{-3}$

19.

$x\ne 1;$ when we solve this we get $x=1,$ which is excluded, therefore NO solution

21.

$x\ne 0;$ $x=-\frac{5}{2}$

23.

$y=-\frac{4}{5}x+\frac{14}{5}$

25.

$y=-\frac{3}{4}x+2$

27.

$y=\frac{1}{2}x+\frac{5}{2}$

29.

$y=\mathrm{-3}x-5$

31.

$y=7$

33.

$y=\mathrm{-4}$

35.

$8x+5y=7$

37.

Parallel

39.

Perpendicular

41.

$m=-\frac{9}{7}$

43.

$m=\frac{3}{2}$

45.

$m_1=-\frac{1}{3},m_2=3;\text{Perpendicular}\text{.}$

47.

$y=0.245x-\mathrm{45.662.}$ Answers may vary. $y_{\text{min}}=\mathrm{-50},y_{\text{max}}=\mathrm{-40}$

49.

$y=-2.333x+\mathrm{6.667.}$ Answers may vary. $y_{\min }=\mathrm{-10}, y_{\max }=10$

51.

$y=-\frac{A}{B}x+\frac{C}{B}$

53.

$\begin{array}{l}\text{The slope for }(\mathrm{-1},1)\text{to }(0,4)\text{is }3.\\ \text{The slope for }(\mathrm{-1},1)\text{to }(2,0)\text{is }\frac{-1}{3}.\\ \text{The slope for }(2,0)\text{to }(3,3)\text{is }3.\\ \text{The slope for }(0,4)\text{to }(3,3)\text{is }\frac{-1}{3}.\end{array}$

Yes they are perpendicular.

55.

30 ft

57.

$57.50

59.

220 mi

2.3 Section Exercises

1.

Answers may vary. Possible answers: We should define in words what our variable is representing. We should declare the variable. A heading.

3.

$2,000-x$

5.

$v+10$

7.

Ann: $23;$ Beth: $46$

9.

$20+0.05m$

11.

300 min

13.

$90+40P$

15.

6 devices

17.

$50,000-x$

19.

4 h

21.

She traveled for 2 h at 20 mi/h, or 40 miles.

23.

$5,000 at 8% and $15,000 at 12%

25.

$B=100+.05x$

27.

Plan A

29.

$R=9$

31.

$r=\frac{4}{5}$ or 0.8

33.

$W=\frac{P-2L}{2}=\frac{58-2(15)}{2}=14$

35.

$f=\frac{pq}{p+q}=\frac{8(13)}{8+13}=\frac{104}{21}$

37.

$m=\frac{-5}{4}$

39.

$h=\frac{2A}{b_1+b_2}$

41.

length = 360 ft; width = 160 ft

43.

405 mi

45.

$A=88\text{in}{.}^2$

47.

28.7

49.

$h=\frac{V}{\pi r^2}$

51.

$r=\sqrt{\frac{V}{\pi h}}$

53.

$C=12\pi$

2.4 Section Exercises

1.

Add the real parts together and the imaginary parts together.

3.

Possible answer: $i$ times $i$ equals -1, which is not imaginary.

5.

$\mathrm{-8}+2i$

7.

$14+7i$

9.

$-\frac{23}{29}+\frac{15}{29}i$

11.

13.

15.

$8-i$

17.

$\mathrm{-11}+4i$

19.

$2\mathrm{-5}i$

21.

$6+15i$

23.

$\mathrm{-16}+32i$

25.

$\mathrm{-4}\mathrm{-7}i$

27.

25

29.

$2-\frac{2}{3}i$

31.

$4-6i$

33.

$\frac{2}{5}+\frac{11}{5}i$

35.

$15i$

37.

$1+i\sqrt{3}$

39.

$1$

41.

$\mathrm{-1}$

43.

128i

45.

${\left(\frac{\sqrt{3}}{2}+\frac{1}{2}i\right)}^6=\mathrm{-1}$

47.

$3i$

49.

0

51.

$5\mathrm{-5}i$

53.

$\mathrm{-2}i$

55.

$\frac{9}{2}-\frac{9}{2}i$

2.5 Section Exercises

1.

It is a second-degree equation (the highest variable exponent is 2).

3.

We want to take advantage of the zero property of multiplication in the fact that if $a\cdot b=0$ then it must follow that each factor separately offers a solution to the product being zero: $a=0or\text{b}=0.$

5.

One, when no linear term is present (no x term), such as $x^2=16.$ Two, when the equation is already in the form ${(ax+b)}^2=d.$

7.

$x=6,$ $x=3$

9.

$x=\frac{-5}{2},$ $x=\frac{-1}{3}$

11.

$x=5,$ $x=\mathrm{-5}$

13.

$x=\frac{-3}{2},$ $x=\frac{3}{2}$

15.

$x=\mathrm{-2},3$

17.

$x=0,$ $x=\frac{-3}{7}$

19.

$x=\mathrm{-6},$ $x=6$

21.

$x=6,$ $x=\mathrm{-4}$

23.

$x=1,$ $x=\mathrm{-2}$

25.

$x=\mathrm{-2},$ $x=11$

27.

$x=3\pm \sqrt{22}$

29.

$z=\frac{2}{3},$ $z=-\frac{1}{2}$

31.

$x=\frac{3\pm \sqrt{17}}{4}$

33.

Not real

35.

One rational

37.

Two real; rational

39.

$x=\frac{-1\pm \sqrt{17}}{2}$

41.

$x=\frac{5\pm \sqrt{13}}{6}$

43.

$x=\frac{-1\pm \sqrt{17}}{8}$

45.

$x\approx 0.131$ and $x\approx 2.535$

47.

$x\approx -6.7$ and $x\approx 1.7$

49.

$\begin{array}{ccc}\hfill a{x}^2+bx+c& =& 0\hfill \\ \hfill x^2+\frac{b}{a}x& =& \frac{-c}{a}\hfill \\ \hfill x^2+\frac{b}{a}x+\frac{b^2}{4a^2}& =& \frac{-c}{a}+\frac{b}{4a^2}\hfill \\ \hfill {\left(x+\frac{b}{2a}\right)}^2& =& \frac{b^2-4ac}{4a^2}\hfill \\ \hfill x+\frac{b}{2a}& =& \pm \sqrt{\frac{b^2-4ac}{4a^2}}\hfill \\ \hfill x& =& \frac{-b\pm \sqrt{b^2-4ac}}{2a}\hfill \end{array}$

51.

$x(x+10)=119;$ 7 ft. and 17 ft.

53.

maximum at $x=70$

55.

The quadratic equation would be $(100x\mathrm{-0.5}{x}^2)-(60x+300)=300.$ The two values of $x$ are 20 and 60.

57.

3 feet

2.6 Section Exercises

1.

This is not a solution to the radical equation, it is a value obtained from squaring both sides and thus changing the signs of an equation which has caused it not to be a solution in the original equation.

3.

They are probably trying to enter negative 9, but taking the square root of $\mathrm{-9}$ is not a real number. The negative sign is in front of this, so your friend should be taking the square root of 9, cubing it, and then putting the negative sign in front, resulting in $\mathrm{-27.}$

5.

A rational exponent is a fraction: the denominator of the fraction is the root or index number and the numerator is the power to which it is raised.

7.

$x=81$

9.

$x=17$

11.

$x=8, x=27$

13.

$x=\mathrm{-2},1,\mathrm{-1}$

15.

$y=0, \frac{3}{2}, \frac{-3}{2}$

17.

$m=1,\mathrm{-1}$

19.

$x=\frac{2}{5},\mathrm{\pm 3}i$

21.

$x=32$

23.

$t=\frac{44}{3}$

25.

$x=3$

27.

$x=\mathrm{-2}$

29.

$x=4,\frac{\mathrm{-4}}{3}$

31.

$x=\frac{-5}{4},\frac{7}{4}$

33.

$x=3,\mathrm{-2}$

35.

$x=\mathrm{-5}$

37.

$x=1,\mathrm{-1},3,-3$

39.

$x=2,\mathrm{-2}$

41.

$x=1,5$

43.

$x\ge 0$

45.

$x=4,6,\mathrm{-6},\mathrm{-8}$

47.

10 in.

49.

90 kg

2.7 Section Exercises

1.

When we divide both sides by a negative it changes the sign of both sides so the sense of the inequality sign changes.

3.

$\left(-\infty ,\infty \right)$

5.

We start by finding the x-intercept, or where the function = 0. Once we have that point, which is $(3,0),$ we graph to the right the straight line graph $y=x\mathrm{-3},$ and then when we draw it to the left we plot positive y values, taking the absolute value of them.

7.

$\left(-\infty ,\frac{3}{4}\right]$

9.

$\left[-\frac{13}{2},\infty \right)$

11.

$\left(-\infty ,3\right)$

13.

$\left(-\infty ,-\frac{37}{3}\right]$

15.

All real numbers $\left(-\infty ,\infty \right)$

17.

$\left(-\infty ,-\frac{10}{3}\right)\cup \left(4,\infty \right)$

19.

$\left(-\infty ,\mathrm{-4}\right]\cup \left[8,+\infty \right)$

21.

No solution

23.

$\left(\mathrm{-5},11\right)$

25.

$\left[6,12\right]$

27.

$\left[\mathrm{-10},12\right]$

29.

$\begin{array}{ll}x>-6\text{and}x>-2\hfill & \text{Take the intersection of two sets}.\hfill \\ x>-2,(-2,+\infty )\hfill & \hfill \end{array}$

31.

$\begin{array}{ll}x<-3\mathrm{or}x\ge 1\hfill & \text{Take the union of the two sets}.\hfill \\ (-\infty ,-3){{{\displaystyle \cup }}^{\text{}}}^{\text{}}[1,\infty )\hfill & \hfill \end{array}$

33.

$\left(-\infty ,\mathrm{-1}\right)\cup \left(3,\infty \right)$

35.

$\left[\mathrm{-11},\mathrm{-3}\right]$

37.

It is never less than zero. No solution.

39.

Where the blue line is above the orange line; point of intersection is $x=-3.$

$\left(-\infty ,\mathrm{-3}\right)$

41.

Where the blue line is above the orange line; always. All real numbers.

$(-\infty ,-\infty )$

43.

$\left(\mathrm{-1},3\right)$

45.

$\left(-\infty ,4\right)$

47.

$\{x|x<6\}$

49.

$\{x|\mathrm{-3}\le x<5\}$

51.

$\left(\mathrm{-2},1\right]$

53.

$\left(-\infty ,4\right]$

55.

Where the blue is below the orange; always. All real numbers. $(-\infty ,+\infty ).$

57.

Where the blue is below the orange; $\left(1,7\right).$

59.

$x=2,\frac{-4}{5}$

61.

$\left(\mathrm{-7},5\right]$

63.

$\begin{array}{l}80\le T\le 120\\ 1,600\le 20T\le 2,400\end{array}$

$\left[1,600,2,400\right]$

Review Exercises

1.

x-intercept: $\left(3,0\right);$ y-intercept: $\left(0,\mathrm{-4}\right)$

3.

$y=\frac{5}{3}x+4$

5.

$\sqrt{72}=6\sqrt{2}$

7.

$620.097$

9.

midpoint is $\left(2,\frac{23}{2}\right)$

11.

x	y
0	−2
3	2
6	6

13.

$x=4$

15.

$x=\frac{12}{7}$

17.

No solution

19.

$y=\frac{1}{6}x+\frac{4}{3}$

21.

$y=\frac{2}{3}x+6$

23.

females 17, males 56

25.

84 mi

27.

$x=-\frac{3}{4}\pm \frac{i\sqrt{47}}{4}$

29.

horizontal component $\mathrm{-2};$ vertical component $\mathrm{-1}$

31.

$7+11i$

33.

$16i$

35.

$\mathrm{-16}-30i$

37.

$\mathrm{-4}-i\sqrt{10}$

39.

$x=7-3i$

41.

$x=\mathrm{-1},\mathrm{-5}$

43.

$x=0,\frac{9}{7}$

45.

$x=10,\mathrm{-2}$

47.

$x=\frac{-1\pm \sqrt{5}}{4}$

49.

$x=\frac{2}{5},\frac{-1}{3}$

51.

$x=5\pm 2\sqrt{7}$

53.

$x=0,256$

55.

$x=0,\pm \sqrt{2}$

57.

$x=\mathrm{-2}$

59.

$x=\frac{11}{2},\frac{\mathrm{-17}}{2}$

61.

$\left(-\infty ,4\right)$

63.

$\left[\frac{-10}{3},2\right]$

65.

No solution

67.

$\left(-\frac{4}{3},\frac{1}{5}\right)$

69.

Where the blue is below the orange line; point of intersection is $x=\mathrm{3.5.}$

$\left(3.5,\infty \right)$

Practice Test

1.

$y=\frac{3}{2}x+2$

x	y
0	2
2	5
4	8

3.

$\left(0,\mathrm{-3}\right)$ $\left(4,0\right)$

5.

$\left(-\infty ,9\right]$

7.

$x=\mathrm{-15}$

9.

$x\ne \mathrm{-4},2;$ $x=\frac{-5}{2},1$

11.

$x=\frac{3\pm \sqrt{3}}{2}$

13.

$\left(\mathrm{-4},1\right)$

15.

$y=\frac{\mathrm{-5}}{9}x-\frac{2}{9}$

17.

$y=\frac{5}{2}x-4$

19.

$14i$

21.

$\frac{5}{13}-\frac{14}{13}i$

23.

$x=2,\frac{-4}{3}$

25.

$x=\frac{1}{2}\pm \frac{\sqrt{2}}{2}$

27.

$4$

29.

$x=\frac{1}{2},2,\mathrm{-2}$