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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

how do I set up the problem?
Harshika Reply
what is a solution set?
Harshika
find the subring of gaussian integers?
Rofiqul
hello, I am happy to help!
Shirley Reply
please can go further on polynomials quadratic
Abdullahi
hi mam
Mark
I need quadratic equation link to Alpa Beta
Abdullahi Reply
find the value of 2x=32
Felix Reply
divide by 2 on each side of the equal sign to solve for x
corri
X=16
Michael
Want to review on complex number 1.What are complex number 2.How to solve complex number problems.
Beyan
yes i wantt to review
Mark
use the y -intercept and slope to sketch the graph of the equation y=6x
Only Reply
how do we prove the quadratic formular
Seidu Reply
please help me prove quadratic formula
Darius
hello, if you have a question about Algebra 2. I may be able to help. I am an Algebra 2 Teacher
Shirley Reply
thank you help me with how to prove the quadratic equation
Seidu
may God blessed u for that. Please I want u to help me in sets.
Opoku
what is math number
Tric Reply
4
Trista
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Sidiki Reply
can you teacch how to solve that🙏
Mark
Solve for the first variable in one of the equations, then substitute the result into the other equation. Point For: (6111,4111,−411)(6111,4111,-411) Equation Form: x=6111,y=4111,z=−411x=6111,y=4111,z=-411
Brenna
(61/11,41/11,−4/11)
Brenna
x=61/11 y=41/11 z=−4/11 x=61/11 y=41/11 z=-4/11
Brenna
Need help solving this problem (2/7)^-2
Simone Reply
x+2y-z=7
Sidiki
what is the coefficient of -4×
Mehri Reply
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
Alfred Reply
An investment account was opened with an initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
Kala Reply
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
Moses Reply
Practice Key Terms 4

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Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
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