Skip to ContentGo to accessibility pageKeyboard shortcuts menu
OpenStax Logo
College Algebra with Corequisite Support

Chapter 2

College Algebra with Corequisite SupportChapter 2

Try It

2.1 The 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 )
This is an image of a graph on an x, y coordinate plane. The x and y-axis range from negative 5 to 5.  A line passes through the points (-2, 1); (-1, 3/2); (0, 2); (1, 5/2); and (2, 3).
2.

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

This is an image of a line graph on an x, y coordinate plane. The x and y axes range from negative 4 to 6.  The function y = -3x/4 + 3 is plotted.
3.

125 =5 5 125 =5 5

4.

( 5, 5 2 ) ( 5, 5 2 )

2.2 Linear 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.

Coordinate plane with the x-axis ranging from negative 5 to 5 and the y-axis ranging from negative 1 to 6.  Two functions are graphed on the same plot: y = x/2 plus 5 and y = x/2 plus 2.  The lines do not cross.
12.

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

2.3 Models 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

2.4 Complex Numbers

1.

−24 =0+2i 6 −24 =0+2i 6

2.
Coordinate plane with the x and y axes ranging from negative 5 to 5.  The point -4  i is plotted.
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

2.5 Quadratic Equations

1.

( x6 )( x+1 )=0;x=6, x=1 ( x6 )( 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

2.6 Other 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.

2.7 Linear 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<x9or( 6,9 ] 6<x9or( 6,9 ]

8.

( 1 8 , 1 2 ) ( 1 8 , 1 2 )

9.

| x−2 |3 | x−2 |3

10.

k1 k1 or k7; k7; in interval notation, this would be (,1][7,). (,1][7,).

A coordinate plane with the x-axis ranging from -1 to 9 and the y-axis ranging from -3 to 8.  The function y = -2|k  4| + 6 is graphed and everything above the function is shaded in.

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=42x y=42x

13.

y= 52x 3 y= 52x 3

15.

y=2x 4 5 y=2x 4 5

17.

d= 74 d= 74

19.

d= 36 =6 d= 36 =6

21.

d62.97 d62.97

23.

( 3, 3 2 ) ( 3, 3 2 )

25.

( 2,−1 ) ( 2,−1 )

27.

( 0,0 ) ( 0,0 )

29.

y=0 y=0

31.


This is an image of an x, y coordinate plane with the x and y axes ranging from negative 5 to 5. The points (0,4); (-1,2) and (2,1) are plotted and labeled.

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
This is an image of an x, y coordinate plane with the x and y axes ranging from negative 10 to 10.  The points (-3, 1); (0, 2); (3, 3) and (6, 4) are plotted and labeled.  A line runs through all these points.
37.
x y
–3 0
0 1.5
3 3
This is an image of an x, y coordinate plane with the x and y axes ranging from negative 10 to 10.  The points (-3, 0); (0, 1.5) and (3, 3) are plotted and labeled.  A line runs through all of these points.
39.
This is an image of an x, y coordinate plane with the x and y axes ranging from negative 10 to 10.  The points (8, 0) and (0, -4) are plotted and labeled.  A line runs through both of these points.
41.
This is an image of an x, y coordinate plane with the x and y axes ranging from negative 10 to 10.  The points (0, 2) and (3, 0) are plotted and labeled.  A line runs through both of these points.
43.

d=8.246 d=8.246

45.

d=5 d=5

47.

( −3,4 ) ( −3,4 )

49.

x=0        y=−2 x=0        y=−2

51.

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

53.

x=1.667y=0 x=1.667y=0

55.

1511.2=3.8mi 1511.2=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.

x1; x1; when we solve this we get x=1, x=1, which is excluded, therefore NO solution

21.

x0; x0; 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=−3x5 y=−3x5

31.

y=7 y=7

33.

y=−4 y=−4

35.

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

37.


Coordinate plane with the x and y axes ranging from negative 10 to 10.  The functions 3 times x minus 2 times y = 5 and 6 times y minus 9 times x = 6 are graphed on the same plot.  The lines do not cross.

Parallel

39.


Coordinate plane with the x and y axes ranging from negative 10 to 10.  The function y = negative 3 and the line x = 4 are graphed on the same plot.  These lines cross at a 90 degree angle.

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=0.245x45.662. y=0.245x45.662. 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. y min =−10,  y max =10 y min =−10,  y max =10

51.

y= A B x+ C B y= A B x+ C B

53.

The slope for (−1,1)to (0,4)is 3. The slope for (−1,1)to (2,0)is  1 3 . The slope for (2,0)to (3,3)is 3. The slope for (0,4)to (3,3)is  1 3 . The slope for (−1,1)to (0,4)is 3. The slope for (−1,1)to (2,0)is  1 3 . The slope for (2,0)to (3,3)is 3. The slope for (0,4)to (3,3)is  1 3 .

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.

2,000x 2,000x

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.

50,000x 50,000x

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= P2L 2 = 582(15) 2 =14 W= P2L 2 = 582(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.


Coordinate plane with the x and y axes ranging from 5 to 5.  The point 1  2i is plotted
13.


Coordinate plane with the x and y axes ranging from -5 to 5.  The point i is plotted.
15.

8i 8i

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.

46i 46i

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 ab=0 ab=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.

x0.131 x0.131 and x2.535 x2.535

47.

x6.7 x6.7 and x1.7 x1.7

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.

He or she is 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.

x=8,  x=27 x=8,  x=27

13.

x=−2,1,−1 x=−2,1,−1

15.

y=0,   3 2 ,   3 2 y=0,   3 2 ,   3 2

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<3orx1 Take the union of the two sets. (,3) [1,) x<3orx1 Take the union of the two sets. (,3) [1,)

33.

( ,−1 )( 3, ) ( ,−1 )( 3, )


A coordinate plane where the x and y axes both range from -10 to 10.  The function |x  1| is graphed and labeled along with the line y = 2.  Along the x-axis there is an open circle at the point -1 with an arrow extending leftward from it.  Also along the x-axis is an open circle at the point 3 with an arrow extending rightward from it.
35.

[ −11,−3 ] [ −11,−3 ]


A coordinate plane with the x-axis ranging from -14 to 10 and the y-axis ranging from -1 to 10.  The function y = |x + 7| and the line y = 4 are graphed.  On the x-axis theres a dot on the points -11 and -3 with a line connecting them.
37.

It is never less than zero. No solution.


A coordinate plane with the x and y axes ranging from -10 to 10.  The function y = |x -2| and the line y = 0 are graphed.
39.

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

( ,−3 ) ( ,−3 )


A coordinate plane with the x and y axes ranging from -10 to 10.  The lines y = x - 2 and y = 2x + 1 are graphed on the same axes.
41.

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

(,) (,)


A coordinate plane with the x and y axes ranging from -10 to 10.  The lines y = x/2 +1 and y = x/2  5 are both graphed on the same axes.
43.

( −1,3 ) ( −1,3 )

45.

( ,4 ) ( ,4 )

47.

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

49.

{ x| −3x<5 } { x| −3x<5 }

51.

( −2,1 ] ( −2,1 ]

53.

( ,4 ] ( ,4 ]

55.

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


A coordinate plane with the x and y axes ranging from -10 to 10.  The function y = -0.5|x + 2| and the line y = 4 are graphed on the same axes.  A line runs along the entire x-axis.
57.

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


A coordinate plane with the x and y axes ranging from -10 to 10.  The function y = |x  4| and the line y = 3 are graphed on the same axes.  Along the x-axis the points 1 and 7 have an open circle around them and a line connects the two.
59.

x=2, 4 5 x=2, 4 5

61.

( −7,5 ] ( −7,5 ]

63.

80T120 1,60020T2,400 80T120 1,60020T2,400

[ 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


A coordinate plane with the x and y axes ranging from -10 to 10.  The points (0,-2); (3,2) and (6,6) are plotted and a line runs through all these points.
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.

−1630i −1630i

37.

−4i 10 −4i 10

39.

x=73i x=73i

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, )


A coordinate plane with the x and y axes ranging from -10 to 10.  The lines y = x + 3 and y = 3x -4 graphed on the same axes.

Practice Test

1.

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

x y
0 2
2 5
4 8


A coordinate plane with the x and y axes ranging from -10 to 10.  The line going through the points (0,2); (2,5); and (4,8) is graphed.
3.

( 0,−3 ) ( 0,−3 ) ( 4,0 ) ( 4,0 )


A coordinate plane with the x and y axes ranging from -10 to 10.  The points (4,0) and (0,-3) are plotted with a line running through them.
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 x4 y= 5 2 x4

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-corequisite-support/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-corequisite-support/pages/1-introduction-to-prerequisites
Citation information

© Dec 8, 2021 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.