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A man has 72 ft. of fencing to put around a rectangular garden. If the length is 3 times the width, find the dimensions of his garden.

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A truck rental is $25 plus $.30/mi. Find out how many miles Ken traveled if his bill was $50.20.

84 mi

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Complex Numbers

For the following exercises, use the quadratic equation to solve.

2 x 2 + 3 x + 7 = 0

x = 3 4 ± i 47 4

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For the following exercises, name the horizontal component and the vertical component.

−2 i

horizontal component −2 ; vertical component −1

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For the following exercises, perform the operations indicated.

( 9 i ) ( 4 7 i )

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( 2 + 3 i ) ( 5 8 i )

7 + 11 i

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( 3 5 i ) 2

−16 30 i

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2 ( 8 5 )

−4 i 10

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Quadratic Equations

For the following exercises, solve the quadratic equation by factoring.

3 x 2 + 18 x + 15 = 0

x = −1 , −5

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  7 x 2 9 x = 0

x = 0 , 9 7

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For the following exercises, solve the quadratic equation by using the square-root property.

( x 4 ) 2 = 36

x = 10 , −2

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For the following exercises, solve the quadratic equation by completing the square.

4 x 2 + 2 x 1 = 0

x = 1 ± 5 4

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For the following exercises, solve the quadratic equation by using the quadratic formula. If the solutions are not real, state No real solution .

15 x 2 x 2 = 0

x = 2 5 , 1 3

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For the following exercises, solve the quadratic equation by the method of your choice.

x 2 = 10 x + 3

x = 5 ± 2 7

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Other Types of Equations

For the following exercises, solve the equations.

x 1 2 4 x 1 4 = 0

x = 0 , 256

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4 x 3 + 8 x 2 9 x 18 = 0

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3 x 5 6 x 3 = 0

x = 0 , ± 2

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3 x + 7 + x + 2 = 1

x = −2

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| 2 x + 3 | 5 = 9

x = 11 2 , −17 2

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Linear Inequalities and Absolute Value Inequalities

For the following exercises, solve the inequality. Write your final answer in interval notation.

2 x + 5 > x 7

( , 4 )

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x 1 3 + x + 2 5 3 5

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| 3 x + 2 | + 1 9

[ 10 3 , 2 ]

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| x 3 | < −4

No solution

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For the following exercises, solve the compound inequality. Write your answer in interval notation.

3 y < 1 2 y < 5 + y

( 4 3 , 1 5 )

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For the following exercises, graph as described.

Graph the absolute value function and graph the constant function. Observe the points of intersection and shade the x -axis representing the solution set to the inequality. Show your graph and write your final answer in interval notation.

| x + 3 | 5

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Graph both straight lines (left-hand side being y1 and right-hand side being y2) on the same axes. Find the point of intersection and solve the inequality by observing where it is true comparing the y -values of the lines. See the interval where the inequality is true.

x + 3 < 3 x 4

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

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Chapter practice test

Graph the following: 2 y = 3 x + 4.

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.

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Find the x- and y -intercepts for the following:

2 x 5 y = 6

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Find the x- and y -intercepts of this equation, and sketch the graph of the line using just the intercepts plotted.

3 x 4 y = 12

( 0 , −3 ) ( 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.

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Find the exact distance between ( 5 , −3 ) and ( 2 , 8 ) . Find the coordinates of the midpoint of the line segment joining the two points.

Write the interval notation for the set of numbers represented by { x | x 9 } .

( , 9 ]

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Solve for x : 5 x + 8 = 3 x 10.

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Solve for x : 3 ( 2 x 5 ) 3 ( x 7 ) = 2 x 9.

x = −15

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Solve for x : x 2 + 1 = 4 x

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Solve for x : 5 x + 4 = 4 + 3 x 2 .

x −4 , 2 ; x = 5 2 , 1

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The perimeter of a triangle is 30 in. The longest side is 2 less than 3 times the shortest side and the other side is 2 more than twice the shortest side. Find the length of each side.

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Solve for x . Write the answer in simplest radical form.

x 2 3 x = −1 2

x = 3 ± 3 2

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Solve: 3 x 8 4.

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Solve: | 2 x + 3 | < 5.

( −4 , 1 )

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Solve: | 3 x 2 | 4.

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For the following exercises, find the equation of the line with the given information.

Passes through the points ( 4 , 2 ) and ( 5 , −3 ) .

y = −5 9 x 2 9

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Has an undefined slope and passes through the point ( 4 , 3 ) .

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Passes through the point ( 2 , 1 ) and is perpendicular to y = 2 5 x + 3.

y = 5 2 x 4

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Add these complex numbers: ( 3 2 i ) + ( 4 i ) .

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Simplify: −4 + 3 −16 .

14 i

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Multiply: 5 i ( 5 3 i ) .

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Divide: 4 i 2 + 3 i .

5 13 14 13 i

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Solve this quadratic equation and write the two complex roots in a + b i form: x 2 4 x + 7 = 0.

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Solve: ( 3 x 1 ) 2 1 = 24.

x = 2 , 4 3

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Solve: x 2 6 x = 13.

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Solve: 4 x 2 4 x 1 = 0

x = 1 2 ± 2 2

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Solve:

x 7 = x 7

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Solve: 2 + 12 2 x = x

4

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Solve: ( x 1 ) 2 3 = 9

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For the following exercises, find the real solutions of each equation by factoring.

2 x 3 x 2 8 x + 4 = 0

x = 1 2 , 2 , −2

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( x + 5 ) 2 3 ( x + 5 ) 4 = 0

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Questions & Answers

what is subgroup
Purshotam Reply
Prove that: (2cos&+1)(2cos&-1)(2cos2&-1)=2cos4&+1
Macmillan Reply
e power cos hyperbolic (x+iy)
Vinay Reply
10y
Michael
tan hyperbolic inverse (x+iy)=alpha +i bita
Payal Reply
prove that cos(π/6-a)*cos(π/3+b)-sin(π/6-a)*sin(π/3+b)=sin(a-b)
Tejas Reply
why {2kπ} union {kπ}={kπ}?
Huy Reply
why is {2kπ} union {kπ}={kπ}? when k belong to integer
Huy
if 9 sin theta + 40 cos theta = 41,prove that:41 cos theta = 41
Trilochan Reply
what is complex numbers
Ayushi Reply
give me treganamentry question
Anshuman Reply
Solve 2cos x + 3sin x = 0.5
shobana Reply
madras university algebra questions papers first year B. SC. maths
Kanniyappan Reply
Hey
Rightspect
hi
chesky
Give me algebra questions
Rightspect
how to send you
Vandna
What does this mean
Michael Reply
cos(x+iy)=cos alpha+isinalpha prove that: sin⁴x=sin²alpha
rajan Reply
cos(x+iy)=cos aplha+i sinalpha prove that: sinh⁴y=sin²alpha
rajan
cos(x+iy)=cos aplha+i sinalpha prove that: sinh⁴y=sin²alpha
rajan
is there any case that you can have a polynomials with a degree of four?
victor
***sscc.edu/home/jdavidso/math/catalog/polynomials/fourth/fourth.html
Oliver
can you solve it step b step
Ching Reply
give me some important question in tregnamentry
Anshuman
what is linear equation with one unknown 2x+5=3
Joan Reply
-4
Joel
x=-4
Joel
x=-1
Joan
I was wrong. I didn't move all constants to the right of the equation.
Joel
x=-1
Cristian
Adityasuman x= - 1
Aditya
y=x+1
gary
x=_1
Daulat
yas. x= -4
Deepak
x=-1
Deepak
2x=3-5 x=-2/2=-1
Rukmini
-1
Bobmorris
Practice Key Terms 4

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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