Skip to ContentGo to accessibility pageKeyboard shortcuts menu
College Algebra 2e

# Chapter 2

## 2.1The Rectangular Coordinate Systems and Graphs

1.
 $x x$ $y= 1 2 x+2 y= 1 2 x+2$ $( x,y ) ( x,y )$ $−2 −2$ $y= 1 2 ( −2 )+2=1 y= 1 2 ( −2 )+2=1$ $( −2,1 ) ( −2,1 )$ $−1 −1$ $y= 1 2 ( −1 )+2= 3 2 y= 1 2 ( −1 )+2= 3 2$ $( −1, 3 2 ) ( −1, 3 2 )$ $0 0$ $y= 1 2 ( 0 )+2=2 y= 1 2 ( 0 )+2=2$ $( 0,2 ) ( 0,2 )$ $1 1$ $y= 1 2 ( 1 )+2= 5 2 y= 1 2 ( 1 )+2= 5 2$ $( 1, 5 2 ) ( 1, 5 2 )$ $2 2$ $y= 1 2 ( 2 )+2=3 y= 1 2 ( 2 )+2=3$ $( 2,3 ) ( 2,3 )$
2.

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

3.

$125 =5 5 125 =5 5$

4.

$( −5, 5 2 ) ( −5, 5 2 )$

## 2.2Linear Equations in One Variable

1.

$x=−5 x=−5$

2.

$x=−3 x=−3$

3.

$x= 10 3 x= 10 3$

4.

$x=1 x=1$

5.

$x=− 7 17 . x=− 7 17 .$ Excluded values are $x=− 1 2 x=− 1 2$ and $x=− 1 3 . x=− 1 3 .$

6.

$x= 1 3 x= 1 3$

7.

$m=− 2 3 m=− 2 3$

8.

$y=4x−3 y=4x−3$

9.

$x+3y=2 x+3y=2$

10.

Horizontal line: $y=2 y=2$

11.

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

12.

$y=5x+3 y=5x+3$

## 2.3Models and Applications

1.

11 and 25

2.

$C=2.5x+3,650 C=2.5x+3,650$

3.

45 mi/h

4.

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

5.

250 ft2

## 2.4Complex Numbers

1.

$−24 =0+2i 6 −24 =0+2i 6$

2.
3.

$(3−4i)−(2+5i)=1−9i (3−4i)−(2+5i)=1−9i$

4.

$5 2 −i 5 2 −i$

5.

$18+i 18+i$

6.

$−3−4i −3−4i$

7.

$−1 −1$

## 2.5Quadratic Equations

1.

$( x−6 )( x+1 )=0;x=6, x=−1 ( x−6 )( x+1 )=0;x=6, x=−1$

2.

$( x−7 )( x+3 )=0, ( x−7 )( x+3 )=0,$ $x=7, x=7,$ $x=−3. x=−3.$

3.

$( x+5 )( x−5 )=0, ( x+5 )( x−5 )=0,$ $x=−5, x=−5,$ $x=5. x=5.$

4.

$( 3x+2 )( 4x+1 )=0, ( 3x+2 )( 4x+1 )=0,$ $x=− 2 3 , x=− 2 3 ,$ $x=− 1 4 x=− 1 4$

5.

$x=0,x=−10,x=−1 x=0,x=−10,x=−1$

6.

$x=4± 5 x=4± 5$

7.

$x=3± 22 x=3± 22$

8.

$x=− 2 3 , x=− 2 3 ,$ $x= 1 3 x= 1 3$

9.

$5 5$ units

## 2.6Other Types of Equations

1.

$1 4 1 4$

2.

$25 25$

3.

${ −1 } { −1 }$

4.

$0, 0,$ $1 2 , 1 2 ,$ $− 1 2 − 1 2$

5.

$1; 1;$ extraneous solution $− 2 9 − 2 9$

6.

$−2; −2;$ extraneous solution $−1 −1$

7.

$−1, −1,$ $3 2 3 2$

8.

$−3,3,−i,i −3,3,−i,i$

9.

$2,12 2,12$

10.

$−1, −1,$ $0 0$ is not a solution.

## 2.7Linear Inequalities and Absolute Value Inequalities

1.

$[ −3,5 ] [ −3,5 ]$

2.

$( −∞,−2 )∪[ 3,∞ ) ( −∞,−2 )∪[ 3,∞ )$

3.

$x<1 x<1$

4.

$x≥−5 x≥−5$

5.

$( 2,∞ ) ( 2,∞ )$

6.

$[ − 3 14 ,∞ ) [ − 3 14 ,∞ )$

7.

$6

8.

$( − 1 8 , 1 2 ) ( − 1 8 , 1 2 )$

9.

$| x−2 |≤3 | x−2 |≤3$

10.

$k≤1 k≤1$ or $k≥7; k≥7;$ in interval notation, this would be $(−∞,1]∪[7,∞). (−∞,1]∪[7,∞).$

## 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 $( 2,0 ) ( 2,0 )$ and the y-intercept is $( 0,6 ). ( 0,6 ).$

7.

The x-intercept is $( 2,0 ) ( 2,0 )$ and the y-intercept is $( 0,−3 ). ( 0,−3 ).$

9.

The x-intercept is $( 3,0 ) ( 3,0 )$ and the y-intercept is $( 0, 9 8 ). ( 0, 9 8 ).$

11.

$y=4−2x y=4−2x$

13.

$y= 5−2x 3 y= 5−2x 3$

15.

$y=2x− 4 5 y=2x− 4 5$

17.

$d= 74 d= 74$

19.

$d= 36 =6 d= 36 =6$

21.

$d≈62.97 d≈62.97$

23.

$( 3, −3 2 ) ( 3, −3 2 )$

25.

$( 2,−1 ) ( 2,−1 )$

27.

$( 0,0 ) ( 0,0 )$

29.

$y=0 y=0$

31.

not collinear

33.

$A: ( −3,2 ),B: ( 1,3 ),C: ( 4,0 ) A: ( −3,2 ),B: ( 1,3 ),C: ( 4,0 )$

35.
 $x x$ $y y$ $−3 −3$ 1 0 2 3 3 6 4
37.
 x y –3 0 0 1.5 3 3
39.
41.
43.

$d=8.246 d=8.246$

45.

$d=5 d=5$

47.

$( −3,4 ) ( −3,4 )$

49.

51.

$x=0.75y=0 x=0.75y=0$

53.

$x=−1.667y=0 x=−1.667y=0$

55.

$15−11.2=3.8mi 15−11.2=3.8mi$ shorter

57.

$6.042 6.042$

59.

Midpoint of each diagonal is the same point $(2,2)(2,2)$. 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 $x x$ 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.

$x=2 x=2$

9.

$x= 2 7 x= 2 7$

11.

$x=6 x=6$

13.

$x=3 x=3$

15.

$x=−14 x=−14$

17.

$x≠−4; x≠−4;$ $x=−3 x=−3$

19.

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

21.

$x≠0; x≠0;$ $x=− 5 2 x=− 5 2$

23.

$y=− 4 5 x+ 14 5 y=− 4 5 x+ 14 5$

25.

$y=− 3 4 x+2 y=− 3 4 x+2$

27.

$y= 1 2 x+ 5 2 y= 1 2 x+ 5 2$

29.

$y=−3x−5 y=−3x−5$

31.

$y=7 y=7$

33.

$y=−4 y=−4$

35.

$8x+5y=7 8x+5y=7$

37.

Parallel

39.

Perpendicular

41.

$m=− 9 7 m=− 9 7$

43.

$m= 3 2 m= 3 2$

45.

$m 1 =− 1 3 , m 2 =3; Perpendicular. m 1 =− 1 3 , m 2 =3; Perpendicular.$

47.

$y=0.245x−45.662. y=0.245x−45.662.$ Answers may vary. $y min =−50, y max =−40 y min =−50, y max =−40$

49.

$y=−2.333x+6.667. y=−2.333x+6.667.$ Answers may vary.

51.

$y=− A B x+ C B y=− A B x+ C B$

53.

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. $2,000−x 2,000−x$ 5. $v+10 v+10$ 7. Ann: $23; 23;$ Beth: $46 46$ 9. $20+0.05m 20+0.05m$ 11. 300 min 13. $90+40P 90+40P$ 15. 6 devices 17. $50,000−x 50,000−x$ 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.

$B=100+.05x B=100+.05x$

27.

Plan A

29.

$R=9 R=9$

31.

$r= 4 5 r= 4 5$ or 0.8

33.

$W= P−2L 2 = 58−2(15) 2 =14 W= P−2L 2 = 58−2(15) 2 =14$

35.

$f= pq p+q = 8(13) 8+13 = 104 21 f= pq p+q = 8(13) 8+13 = 104 21$

37.

$m= −5 4 m= −5 4$

39.

$h= 2A b 1 + b 2 h= 2A b 1 + b 2$

41.

length = 360 ft; width = 160 ft

43.

405 mi

45.

$A=88in . 2 A=88in . 2$

47.

28.7

49.

$h= V π r 2 h= V π r 2$

51.

$r= V πh r= V πh$

53.

$C=12π C=12π$

## 2.4 Section Exercises

1.

Add the real parts together and the imaginary parts together.

3.

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

5.

$−8+2i −8+2i$

7.

$14+7i 14+7i$

9.

$− 23 29 + 15 29 i − 23 29 + 15 29 i$

11.

13.

15.

$8−i 8−i$

17.

$−11+4i −11+4i$

19.

$2−5i 2−5i$

21.

$6+15i 6+15i$

23.

$−16+32i −16+32i$

25.

$−4−7i −4−7i$

27.

25

29.

$2− 2 3 i 2− 2 3 i$

31.

$4−6i 4−6i$

33.

$2 5 + 11 5 i 2 5 + 11 5 i$

35.

$15i 15i$

37.

$1+i 3 1+i 3$

39.

$1 1$

41.

$−1 −1$

43.

128i

45.

$( 3 2 + 1 2 i ) 6 =−1 ( 3 2 + 1 2 i ) 6 =−1$

47.

$3i 3i$

49.

0

51.

$5−5i 5−5i$

53.

$−2i −2i$

55.

$9 2 − 9 2 i 9 2 − 9 2 i$

## 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 $a⋅b=0 a⋅b=0$ then it must follow that each factor separately offers a solution to the product being zero: $a=0orb=0. a=0orb=0.$

5.

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

7.

$x=6, x=6,$ $x=3 x=3$

9.

$x= −5 2 , x= −5 2 ,$ $x= −1 3 x= −1 3$

11.

$x=5, x=5,$ $x=−5 x=−5$

13.

$x= −3 2 , x= −3 2 ,$ $x= 3 2 x= 3 2$

15.

$x=−2,3 x=−2,3$

17.

$x=0, x=0,$ $x= −3 7 x= −3 7$

19.

$x=−6, x=−6,$ $x=6 x=6$

21.

$x=6, x=6,$ $x=−4 x=−4$

23.

$x=1, x=1,$ $x=−2 x=−2$

25.

$x=−2, x=−2,$ $x=11 x=11$

27.

$x=3± 22 x=3± 22$

29.

$z= 2 3 , z= 2 3 ,$ $z=− 1 2 z=− 1 2$

31.

$x= 3± 17 4 x= 3± 17 4$

33.

Not real

35.

One rational

37.

Two real; rational

39.

$x= −1± 17 2 x= −1± 17 2$

41.

$x= 5± 13 6 x= 5± 13 6$

43.

$x= −1± 17 8 x= −1± 17 8$

45.

$x≈0.131 x≈0.131$ and $x≈2.535 x≈2.535$

47.

$x≈−6.7 x≈−6.7$ and $x≈1.7 x≈1.7$

49.

$a x 2 +bx+c = 0 x 2 + b a x = −c a x 2 + b a x+ b 2 4 a 2 = −c a + b 4 a 2 ( x+ b 2a ) 2 = b 2 −4ac 4 a 2 x+ b 2a = ± b 2 −4ac 4 a 2 x = −b± b 2 −4ac 2a a x 2 +bx+c = 0 x 2 + b a x = −c a x 2 + b a x+ b 2 4 a 2 = −c a + b 4 a 2 ( x+ b 2a ) 2 = b 2 −4ac 4 a 2 x+ b 2a = ± b 2 −4ac 4 a 2 x = −b± b 2 −4ac 2a$

51.

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

53.

maximum at $x=70 x=70$

55.

The quadratic equation would be $(100x−0.5 x 2 )−(60x+300)=300. (100x−0.5 x 2 )−(60x+300)=300.$ The two values of $x x$ 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 $−9 −9$ 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 $−27. −27.$

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.

$x=81 x=81$

9.

$x=17 x=17$

11.

13.

$x=−2,1,−1 x=−2,1,−1$

15.

17.

$m=1,−1 m=1,−1$

19.

$x= 2 5 , ±3 i x= 2 5 , ±3 i$

21.

$x=32 x=32$

23.

$t= 44 3 t= 44 3$

25.

$x=3 x=3$

27.

$x=−2 x=−2$

29.

$x=4, −4 3 x=4, −4 3$

31.

$x= −5 4 , 7 4 x= −5 4 , 7 4$

33.

$x=3,−2 x=3,−2$

35.

$x=−5 x=−5$

37.

$x=1,−1,3,-3 x=1,−1,3,-3$

39.

$x=2,−2 x=2,−2$

41.

$x=1,5 x=1,5$

43.

$x ≥ 0 x ≥ 0$

45.

$x=4,6,−6,−8 x=4,6,−6,−8$

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.

$( −∞,∞ ) ( −∞,∞ )$

5.

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

7.

$( −∞, 3 4 ] ( −∞, 3 4 ]$

9.

$[ − 13 2 ,∞ ) [ − 13 2 ,∞ )$

11.

$( −∞,3 ) ( −∞,3 )$

13.

$( −∞,− 37 3 ] ( −∞,− 37 3 ]$

15.

All real numbers $( −∞,∞ ) ( −∞,∞ )$

17.

$( −∞,− 10 3 )∪( 4,∞ ) ( −∞,− 10 3 )∪( 4,∞ )$

19.

$( −∞,−4 ]∪[ 8,+∞ ) ( −∞,−4 ]∪[ 8,+∞ )$

21.

No solution

23.

$( −5,11 ) ( −5,11 )$

25.

$[ 6,12 ] [ 6,12 ]$

27.

$[ −10,12 ] [ −10,12 ]$

29.

$x>−6andx>−2 Take the intersection of two sets. x>−2, (−2,+∞) x>−6andx>−2 Take the intersection of two sets. x>−2, (−2,+∞)$

31.

$x<−3 or x≥1 Take the union of the two sets. (−∞,−3) ∪ ​ ​ [1,∞) x<−3 or x≥1 Take the union of the two sets. (−∞,−3) ∪ ​ ​ [1,∞)$

33.

$( −∞,−1 )∪( 3,∞ ) ( −∞,−1 )∪( 3,∞ )$

35.

$[ −11,−3 ] [ −11,−3 ]$

37.

It is never less than zero. No solution.

39.

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

$( −∞,−3 ) ( −∞,−3 )$

41.

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

$(−∞,−∞) (−∞,−∞)$

43.

$( −1,3 ) ( −1,3 )$

45.

$( −∞,4 ) ( −∞,4 )$

47.

${ x| x<6 } { x| x<6 }$

49.

${ x| −3≤x<5 } { x| −3≤x<5 }$

51.

$( −2,1 ] ( −2,1 ]$

53.

$( −∞,4 ] ( −∞,4 ]$

55.

Where the blue is below the orange; always. All real numbers. $(−∞,+∞). (−∞,+∞).$

57.

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

59.

$x=2, −4 5 x=2, −4 5$

61.

$( −7,5 ] ( −7,5 ]$

63.

$80≤T≤120 1,600≤20T≤2,400 80≤T≤120 1,600≤20T≤2,400$

$[ 1,600, 2,400 ] [ 1,600, 2,400 ]$

## Review Exercises

1.

x-intercept: $( 3,0 ); ( 3,0 );$ y-intercept: $( 0,−4 ) ( 0,−4 )$

3.

$y= 5 3 x+4 y= 5 3 x+4$

5.

$72 =6 2 72 =6 2$

7.

$620.097 620.097$

9.

midpoint is $( 2, 23 2 ) ( 2, 23 2 )$

11.
 x y 0 −2 3 2 6 6

13.

$x=4 x=4$

15.

$x= 12 7 x= 12 7$

17.

No solution

19.

$y= 1 6 x+ 4 3 y= 1 6 x+ 4 3$

21.

$y= 2 3 x+6 y= 2 3 x+6$

23.

females 17, males 56

25.

84 mi

27.

$x=− 3 4 ± i 47 4 x=− 3 4 ± i 47 4$

29.

horizontal component $−2; −2;$ vertical component $−1 −1$

31.

$7+11i 7+11i$

33.

$16i 16i$

35.

$−16−30i −16−30i$

37.

$−4−i 10 −4−i 10$

39.

$x=7−3i x=7−3i$

41.

$x=−1,−5 x=−1,−5$

43.

$x=0, 9 7 x=0, 9 7$

45.

$x=10,−2 x=10,−2$

47.

$x= −1± 5 4 x= −1± 5 4$

49.

$x= 2 5 , −1 3 x= 2 5 , −1 3$

51.

$x=5±2 7 x=5±2 7$

53.

$x=0,256 x=0,256$

55.

$x=0,± 2 x=0,± 2$

57.

$x=−2 x=−2$

59.

$x= 11 2 , −17 2 x= 11 2 , −17 2$

61.

$( −∞,4 ) ( −∞,4 )$

63.

$[ −10 3 ,2 ] [ −10 3 ,2 ]$

65.

No solution

67.

$( − 4 3 , 1 5 ) ( − 4 3 , 1 5 )$

69.

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

$( 3.5,∞ ) ( 3.5,∞ )$

## Practice Test

1.

$y= 3 2 x+2 y= 3 2 x+2$

x y
0 2
2 5
4 8

3.

$( 0,−3 ) ( 0,−3 )$ $( 4,0 ) ( 4,0 )$

5.

$( −∞,9 ] ( −∞,9 ]$

7.

$x=−15 x=−15$

9.

$x≠−4,2; x≠−4,2;$ $x= −5 2 ,1 x= −5 2 ,1$

11.

$x= 3± 3 2 x= 3± 3 2$

13.

$( −4,1 ) ( −4,1 )$

15.

$y= −5 9 x− 2 9 y= −5 9 x− 2 9$

17.

$y= 5 2 x−4 y= 5 2 x−4$

19.

$14i 14i$

21.

$5 13 − 14 13 i 5 13 − 14 13 i$

23.

$x=2, −4 3 x=2, −4 3$

25.

$x= 1 2 ± 2 2 x= 1 2 ± 2 2$

27.

$4 4$

29.

$x= 1 2 ,2,−2 x= 1 2 ,2,−2$

Citation/Attribution

This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission.

Want to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute OpenStax.

Attribution information
• If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:
Access for free at https://openstax.org/books/college-algebra-2e/pages/1-introduction-to-prerequisites
• If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution:
Access for free at https://openstax.org/books/college-algebra-2e/pages/1-introduction-to-prerequisites
Citation information

© Jun 28, 2024 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.