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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

the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
Kc Reply
1+cos²A/cos²A=2cosec²A-1
Ramesh Reply
test for convergence the series 1+x/2+2!/9x3
success Reply
a man walks up 200 meters along a straight road whose inclination is 30 degree.How high above the starting level is he?
Lhorren Reply
100 meters
Kuldeep
Find that number sum and product of all the divisors of 360
jancy Reply
answer
Ajith
exponential series
Naveen
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
Please you teach
Dua
Yes
ahmed
Thank you
Dua
give me treganamentry question
Anshuman Reply
Solve 2cos x + 3sin x = 0.5
shobana Reply
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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