untitled

Solving Equations and Inequalities

Objective: Solve multi-step equations and inequalities

EXAMPLE

x + 8 = -4

Fill in the Blank

Solve: c + 7 = -5

EXAMPLE

3x = 129

Fill in the Blank

Solve: 14 = 7x

EXAMPLE

2x - 6 = -24

Fill in the Blank

Solve: -5d - 2 = -27

EXAMPLE

5x + 7x + 4 = 28

Fill in the Blank

Solve: -17q + 7q - 13 = 27

​EXAMPLE

​3x - 7 = 5x + 4

Fill in the Blank

8x - 8 = -4x + 16

​Solving Inequalities

​Same steps with ONE EXTRA RULE

​EXAMPLE

​5 - 2x > 11

Multiple Choice

Solve for x:

$$-4x-7\ge21$$  

• $$x\ge7$$  

• $$x\le8$$  

• $$x\ge-7$$  

• $$x\le-7$$  

Solving absolute value equations and inequalities

Review

Solve the absolute value equations

Multiple Choice

What does absolute value mean?

• Everything is positive

• Everything is negative

• Distance away from zero

Multiple Choice

Solve

 $\left|x+5\right|=8$  

• x=6

• x=3

• x=6 or x=2

• x=3 or x=-13

Multiple Choice

Solve

 $\left|x+7\right|=9$  

• x=2 or x= -16

• x=-16

• x=16 

• x=2

Multiple Choice

Solve

 $\left|x-4\right|=5$  

• x= -4  or  x=4

• x=-1 or x = 9

• x= 6 or x= -6

• x=-4 or x=5

Multiple Choice

Solve

 $\left|\frac{x}{10}\right|=2$  

• x= 12

• x=0  or x=12

• x= -20  or x=20

• x= -8

Multiple Choice

Solve

 $\left|2x+4\right|=8$  

• x= 10 or x=-9

• x=2  or x= -6

• x= -4  or  x=6

• x= -2

Solve absolute value inequalities

​

Multiple Choice

Solve the absolute value inequality:

 $\left|x+7\right|>10$  

• -17 < x < 3

• x < -17  or  x > 3

• x > 3

• x < -17

Multiple Choice

Solve the absolute value inequality:

 $\left|x+2\right|\ge3$  

•  $x\le-5\ \ or\ \ \ x\ge1$  
  

•  $x\le-5$  
  

•  $x>1$  
  

•  $-5<x<1$  
  

Multiple Choice

Solve the absolute value inequality:

 $\left|x+2\right|>1$  

•  $x\le-1\ \ or\ \ \ x\ge-3$  
  

•  $x\le-5$  
  

•  $x<-3\ \ or\ \ \ x>-1$  
  

•  $-5<x<1$  
  

Multiple Choice

Solve the absolute value inequality:

 $\left|x-7\right|<6$  

•  $-10<x<-4$  
  

•  $-12<x<-2$  
  

•  $1\le x\le13$  
  

•  $1<x<13$  
  

Multiple Choice

Solve the absolute value inequality:

 $\left|x+1\right|<1$  

•  $0<x<2$  
  

•  $-2<x<0$  
  

•  $-2\le x\le0$  
  

Multiple Choice

Solve the absolute value inequality:

 $\left|2x-5\right|<23$  

•  $9<x\le-14$  
  

•  $-14<x<-9$  
  

•  $-9\le x\le14$  
  

Multiple Choice

Select the graph for the following inequality

 $1\le x\le9$  

Multiple Choice

Select the graph for the following inequality

 $x<-6\ \ \ or\ \ \ x>2$  

Writing Equations and Inequalities

​

Multiple Choice

1.

• A

• B

• C

• D

Multiple Choice

2.

• F

• G

• H

• J

Multiple Choice

3.

• A

• B

• C

• D

Multiple Choice

4.

• F

• G

• H

• J

Multiple Choice

5.

• A

• B

• C

• D

Multiple Choice

6.

• F

• G

• H

• J

Multiple Choice

7.

• A

• B

• C

• D

Multiple Choice

8.

• F

• G

• H

• J

EQUATIONS AND INEQUALITIES

​

Open Ended

BRAIN DUMP!!

WHAT DO YOU REMEMBER ABOUT 2-STEP EQUATIONS ?

TWO-STEP EQUATIONS

HOW TO SOLVE 2-STEP EQUATIONS

​

Multiple Choice

LET'S TRY ONE!!

2X+4=10

WHAT IS THE FIRST STEP WE WILL DO?

• SUBTRACT 4 FROM BOTH SIDES

• ADD 4 TO BOTH SIDES

• DIVIDE BY 4 ON BOTH SIDES

Multiple Choice

LET'S TRY ONE!!

2X+4=10

WHAT IS THE SECOND STEP THAT WE WILL DO ?

• SUBTRACT 2 FROM BOTH SIDES

• ADD 2 TO BOTH SIDES

• DIVIDE BY 2 ON BOTH SIDES

Multiple Choice

LET'S TRY ONE!!

2X+4=10

WHAT IS THE ANSWER?

• X=7

• X=3

• X=10

Open Ended

TRY THIS ONE ON YOUR OWN ?

3X +17= 2

PRACTICE PROBLEMS

PLEASE COMPLETE EACH PRACTICE PROBLEM

Multiple Choice

PLEASE ANSWER THE QUESTION

• X=0

• X=9

• X=19

• X=7

Multiple Choice

PLEASE ANSWER THE QUESTION

• X=0

• X=9

• X=1

• X=7

INEQUALITIES

SOLVE THE EXACT SAME WAY YOU WOULD SOLVE EQUATIONS... JUST FLIP THE SIGN WHEN YOU DIVIDE BY A NEGATIVE NUMBER?

Multiple Choice

 $-2X\ +10<11$  

DO WE FLIP THE SIGN ?

• YES

• NO

Open Ended

 $-2X\ +10<11$  

WHY DO WE FLIP THE SIGN ?

Multiple Choice

 $-2X\ +10<12$  

SOLVE THE INEQUALITY?

•  $X<-1$  

•  $X>-1$  

•  $X\ge-1$  

•  $X\le-1$  

Poll

WHAT DOES THIS SYMBOL REPRESENT?

>

• GREATER LAN SYMBOL

• LESS THAN SYMBOL

• LESS THAN OR EQUAL

• GREATER THAN OR EQUAL

Poll

 $\ge$  

WHAT DOES THIS SYMBOL REPRESENT?

<

• GREATER LAN SYMBOL

• LESS THAN SYMBOL

• LESS THAN OR EQUAL

• GREATER THAN OR EQUAL

Poll

WHAT DOES THIS SYMBOL REPRESENT?



 $\le$  

• GREATER LAN SYMBOL

• LESS THAN SYMBOL

• LESS THAN OR EQUAL

• GREATER THAN OR EQUAL

INEQUALITY SYMBOL MEANINGS

REMEMBER THAT EACH SYMBOL HAS WORDS THAT REPRESENT THE MEANING

LET'S SOLVE SOME

Multiple Choice

please answer the question correctly

• F

• G

• H

• J

Multiple Choice

PLEASE ANSWER THE QUESTION CORRECTLY

• F

• G

• H

• J

Equations and Inequalities TEST REVIEW

Unit 3.2/4.1 Test Review

. 

Multiple Choice

Which of the following is not an equation?

• 2 + 2 = 4

• 2x + 4 = 25

• 3(4 + 5)

• 10 - 2x = 4

Multiple Choice

Which of the following is an equation?

• 5 + 8 = 13

• 2x + 3

• 5 + 9

• 3 + x

Multiple Choice

• A

• B

• C

• D

Multiple Choice

Which of these can be written as an equation?

• Two times 0.75 plus m

• Three less than twice a

• Half the product of five and j

• Four times n is 24

Multiple Choice

Which of the following is an expression?

• 2x = 4

• 5 + 7

• 5 + x = 13

• 3 + 6 = 9

Multiple Choice

• A

• B

• C

• D

Solving Equations and Inequalities

Your main goal is to isolate the variable.

Remember:

multiplication <=> division

addition <=> subtraction

Multiple Choice

• F

• G

• H

• J

Multiple Choice

y + 51 = 99

• y =140

• y =150

• y=48

• Answer not Given

Multiple Choice

What equation is modeled here?

• x - 6 = 13
• x + 6 = 12
• x + 6 = 13
• x6 = 13

Multiple Choice

5.    6b = 30

• x = 36
• x = 24
• x = 180
• x = 5

Multiple Choice

• A

• B

• C

• D

Multiple Choice

The inverse operation of addition is _______________.

• Division
• Multiplication
• Addition
• Subtraction

Multiple Choice

x - 62 = 123
How should you show your work to isolate the variable and solve the equation?

• Add 62 to each side. 
• Subtract 62 from each side.
• Multiply each side by 62.
• Add 123 to each side.

Multiple Choice

Write an equation for the model above.

• x + 3 = -9

• x - 3 = 9

• 3x + 1 = 9

• 3x - 1 = 9

Multiple Choice

x + 12 = 20

• x = 32
• x = -32
• x = 8
• x = -8

Inequalities

Use elimination strategy to find the correct answer.

Multiple Choice

Pick the correct letter for

x < 6?

• A

• B

• C

• D

Multiple Choice

Match the graph with its inequality.

• a > 11

• a < 11

• a ≤ 11

• a ≥ 11

Multiple Choice

Which inequality is represented by the following graph

• x < -8

• x > -8

• x ≥ -8

• x ≤ -8

Multiple Choice

What inequality does the number line graph represent?

• x ≤ 4

• x ≥ -4

• x < -4

• x < 4

Functions

Today's Objective: I can determine if a relation is a function

Functions

• When the x that you try gives you only one y THAT'S A FUNCTION

• A function is a special relation where each x-value only has one y-value

When you put an input into a machine you can only get out one input.

THAT'S A FUNCTION!

(Each input has only one output)

These are all functions

EACH X VALUE

HAS ONLY ONE Y VALUE

These are NOT functions

In each example

there is an X VALUE that has more than one Y VALUE

Multiple Choice

Is the relation a function? Why.

• Yes, because the x-value 11 has two y-values paired with it.

• Yes, because each x-value has only one y-value paired with it.

• No, because the x-value 11 has two y-values paired with it.

• No, because each x-value has only one y-value paired with it.

Multiple Choice

Is this table a function or not a function?

• Function

• Not a Function

Multiple Choice

Is this mapping a function or not a function?

• Function

• Not a Function

Multiple Choice

Is this set of ordered pairs a function or not a function?

• Function

• Not a Function

Multiple Choice

Is this set of ordered pairs a function or not a function?

• Function

• Not a Function

The vertical line test

• To tell if a graph is a function, draw a vertical line anywhere on the graph.

• If any vertical line touches the graph in more than one location it is NOT a function

Multiple Choice

Which graph does NOT pass the vertical line test?

• Graph 1

• Graph 2

• Graph 3

• Graph 4

Multiple Choice

Is this graph a function or not a function?

• Function

• Not a Function

Multiple Choice

Is this graph a function or not a function?

• Function

• Not a Function

Multiple Choice

Is this graph a function or not a function?

• Function

• Not a Function

Multiple Choice

Is this graph a function or not a function?

• Function

• Not a Function

Multiple Choice

Is this graph a function or not a function?

• Function

• Not a Function

Quadratic Functions

Graphs of Quadratic Functions

• Is called a parabola

• If the coefficient of x² is positive, the parabola open upward

• If the coefficient of x² is negative the parabola opens downward

Multiple Choice

What would be an example of a quadratic equation?

• x+5

• x³+2x²+5x-2

• x²+3x-10

Fill in the Blank

The graph of any quadratic function is called

The vertex (or turning point) of the parabola is the lowest point on the graph when it opens upward and the highest when it opens downward.

​

Multiple Choice

is the lowest or highest point of the parabola

• vertex

• axis

• equation

• symmetry

Multiple Choice

Does this graph in the back have maximum or minimum value?

• maximum
• minimum
• neither
• both

The standard form of a quadratic equation

f(x)=a(x-h)²+k, where the vertex is (h,k)

If a is positive, the parabola open upward but if a is negative it opens downward

Multiple Choice

In the function f(x)=2(x+5)²-8, the parabola opens

• upward

• downward

Multiple Choice

In the function f(x)=-5(x+3)²-8, the parabola opens:

• upward

• downward

Finding the vertex of a parabola in standard form

In the function f(x)=(x+1)²-2, the vertex is (-1,-2)

*change the sign of h, k remains the same

In the function f(x)=-2(x-1)²+3, the vertex is (1,3)

• change the sign of h

• the sign of k remain the same

Multiple Choice

If given the equation
y = 3(x + 5)² - 4, what is the vertex of the parabola?

• (5, -4)
• (-5, -4)
• (-15, -4)
• (15, -4)

Multiple Choice

Identify the vertex and whether the graph opens up or down.

• (-1, 4); opens up
• (-1, 4); opens down
• (1, 4); opens up
• (1, 4); opens down

Trigonometric functions graphs

​

What is a function?

A set of ordered pairs (x,y), with only one "y" for an "x" ....

Sine and cosine functions are described by

• Amplitude

• Period

• Phase shift

• Vertical shift

Multiple Choice

What is the amplitude in this graph?

• 1

• 2

• 3

Multiple Choice

What is the period in this graph?

• 1

• $\pi$

• 2 $\pi$

Domain and Range

By Jason Dunn

D = {-3, -1, 1, 4, 5}​

Order doesn't actually matter, but often you'll see least to greatest.

Domain is the set of X values

R = {-5, 0, 2, 6}

​

Don't repeat the 2​

​

Range is the set of Y values

Open Ended

What is the domain of this chart of values?

Domain (X) = {-1, 0, 1, 2, 3}

​

Range (Y) = {1, 2, 3}​

​

​Can't remember which is which? D and R are alphabetical order and so are X and Y.

"Discrete" Function is scattered points​

​

Multiple Choice

Which of these sets is the RANGE of the graph?

• {0, 6, 12, 18}

• {-2, 0, 2, 4}

• {-2, 2, 4}

• {6, 12, 18}

Multiple Choice

Which number is not an element of the domain?

• 2

• 1

• 0

• 6

Fill in the Blank

What is the domain when the range is 3?

R: {-4 < y < 0}

​Graph goes as low as -4 and as high as 0

"Continuous" Graph

​

D: {-3 < x < 1}

​Graph goes as far left as -3, and as far right as 1

​

Multiple Choice

What is the domain of this graph?

• {-2 < x < 4}

• {-5 < x < 4}

• {-4, 0, -2}

• {-4 < x < 2}

Multiple Choice

Which number is outside the domain for this zig-zag shaped graph?

• -3

• 2

• 0

• 4

Open Ended

What is the range of this circle?

Some graphs go forever in a direction (just one inequality sign)

Multiple Choice

What is the domain of this graph?

• All Real Numbers

• x > -2

• x < 6

• -2 < x < 6

Domain is "all real numbers"

​

It looks like it goes full vertical, but it doesn't. Diagonal forever!​

​

Range is "everything above 0" so {y > 0}

Memorize

Multiple Choice

Parabolas go up forever. What is the range?

• y > -2

• y > 0

• y < 5

• y < 2

Multiple Choice

What is the range of this graph?

• y < 3

• y > -3

• y > 0

• All Real Numbers

Domain and Range

Multiple Choice

What variable typically represents the domain?

• x

• y

Multiple Choice

What variable typically represents the range?

• x

• y

Discrete Function

The points are separated.

List the x-values

List the y-values

This is a continuous function.

Use inequalities to list domain and range.

Example 1:

Domain:__________

Range:___________

Multiple Choice

What is the RANGE of this graph?

• -7<x<5

• -7≤y<5

• -3<x<1

• -3≤y<1

Multiple Choice

What is the DOMAIN of this graph?

• -4 ≤ x ≤ 3

• -1 ≤ x ≤ 4

• 4 < x < 3

• -1 < x < 4

Ex. 2

Domain:___________

Range:___________

Multiple Choice

What is the range of the following graph?

• -5, -1,  0 , 1, 5
• -4, -2, 0, 3, 6
• (-4, 1)  ( 6, 0)
• (6, 0)

Finding Domain and Range with other representations.

Mapping

Table

List of ordered pairs

Multiple Choice

What is the domain?

• {5, 10, 15}

• {(5,2), (5,4), (10,6), (15,8)}

• {2, 4, 6, 8}

• {5, 2, 4, 10, 15}

Multiple Choice

What is the range of the mapping?

• {1, 2, 4}

• {0, 1, 2, 3}

• {0, 3}

• {1, 4}

Multiple Choice

Identify the domain of the following relation:
Remember domain is the "x" values

• -6, 14, 2, 28
• (-6,14), (0,32), (2,38), (4,44)
• -6, 0, 2, 4
• 14, 32, 38, 44

Multiple Choice

For the function {(0,1), (1,-3), (2,-4), (-4,1)}, write the domain and range.

• D: {1, -3, -4,}
  R: {0, 1, 2, -4}
• D: {-4, 0, 1, 2}
  R:{-4, -3, 1}
• D: {0, 1, 2, 3, 4}
  R:{1, -3, -4}
• D: {0, 1, 2, -4}
  R: {1, -3, -4, 1}

Function Notation

Find range of the function if the domain is {1, 3, 5}

Multiple Choice

Evaluate f(-2) if f(x) = x²+ 3.

• -1
• 1
• 7
• -7

Solving Logarithmic Equations

​

Review

Solving Log Equations

Things to keep in mind:

-Are the bases of the logs the same?

- Is there any property we can use to simplify the equation?

Example 1

Multiple Choice

Solve  $\log_2\left(x+8\right)=\log_264$  

• x=16

• x=24

• x=56

• x=40

Multiple Choice

Solve $\log_8\left(3x+7\right)=\log_8\left(7x+4\right)$  

•  $x=\frac{3}{4}$  

•  $x=3$  

•  $x=6$   

•  $x=\frac{4}{3}$  

Example 2

Multiple Choice

Solve  $\log_6x+\log_69=\log_654$  

• x=6

• x=45

• x=7

• x=36

Example 3

Multiple Choice

Solve

   $\log_7n=\frac{2}{3}\log_78$  

• n=1

• n=3

• n=6

• n=4

Example 4

Multiple Choice

Solve $\log_9\left(3u+14\right)-\log_95=\log_92u$  

• u=4

• u=6

• u=2

• u=1

Poll

Which of the following best describes your understanding of this lesson?

• Mastered

• Good understanding

• Need more practice

• I don't understand

Solving Logarithmic Equations

​

Solving With Logs of BOTH Sides of =

• Condense logs, if possible (Addition to Multiplication, Subtraction to Division)

• Logs cancel each other out

• Solve the remaining equation

Multiple Choice

Solve log₂(2x - 3) = log₂(4x + 5)

• 4

• -4

• -.33

• .33

Multiple Choice

Solve: $\log_7\left(28+3r\right)=\log_7\left(10r\right)$  

• 4

• 7

• 20

• 28

Multiple Choice

Solve: log₃ (5) + log₃(x) = log₃(2) + log₃(x + 9)

• 5

• 11

• 6

• 3

Solving with Logs on ONE Side of =

• Condense logs, if possible (Addition to Multiplication, Subtraction to Division)

• Isolate the log term (Add/Subtract, Divide by Coefficient)

• Change to Exponential Form

• Solve the Remaining Equation

Multiple Choice

log₆(2x + 3) = 3

• x = 106.5

• x = 100

• x = 16

• x = 50

Multiple Choice

log₃ (x-3) + 10 = 14

• 5

• 13

• 84

• -20

Multiple Choice

log₂(2) + log₂(8x) = 6

• x = 3
• x = 2
• x = 6
• x = 4

Solving Exponentials for Missing Exponents

• Isolate the exponential term (Add/Subtract, Divide by Coefficient)

• Change to logarithmic form

• Solve remaining equation (Change of Base Formula)

Multiple Choice

3^{2x – 6}  = 81

• x = log 4
• x = 5
• x = 4
• x = -1

Multiple Choice

7ⁿ⁺¹⁰- 8 = 6

• -7.374
• -8.643
• -7.360
• -8.853

Solving Logarithmic Equations

​

Solving With Logs of BOTH Sides of =

• Condense logs, if possible (Addition to Multiplication, Subtraction to Division)

• Logs cancel each other out

• Solve the remaining equation

Multiple Choice

Solve log₂(2x - 3) = log₂(4x + 5)

• 4

• -4

• -.33

• .33

Multiple Choice

Solve: $\log_7\left(28+3r\right)=\log_7\left(10r\right)$  

• 4

• 7

• 20

• 28

Multiple Choice

Solve: log₃ (5) + log₃(x) = log₃(2) + log₃(x + 9)

• 5

• 11

• 6

• 3

Solving with Logs on ONE Side of =

• Condense logs, if possible (Addition to Multiplication, Subtraction to Division)

• Isolate the log term (Add/Subtract, Divide by Coefficient)

• Change to Exponential Form

• Solve the Remaining Equation

Multiple Choice

log₆(2x + 3) = 3

• x = 106.5

• x = 100

• x = 16

• x = 50

Multiple Choice

log₃ (x-3) + 10 = 14

• 5

• 13

• 84

• -20

Multiple Choice

log₂(2) + log₂(8x) = 6

• x = 3
• x = 2
• x = 6
• x = 4

Solving Exponentials for Missing Exponents

• Isolate the exponential term (Add/Subtract, Divide by Coefficient)

• Change to logarithmic form

• Solve remaining equation (Change of Base Formula)

Multiple Choice

3^{2x – 6}  = 81

• x = log 4
• x = 5
• x = 4
• x = -1

Multiple Choice

7ⁿ⁺¹⁰- 8 = 6

• -7.374
• -8.643
• -7.360
• -8.853

Real Numbers

Algebra 2

Multiple Choice

The number is a/an...

• Rational Number

• Irrational Number

• Integer

• Whole Number

Multiple Choice

A repeating decimal is what type of number?

• Rational

• Irrational

Multiple Choice

The number is a/an...

• Natural Number

• Irrational Number

• Integer

• Whole Number

Multiple Choice

The number is NOT a/an...

• Natural Number

• Irrational Number

• Integer

• Whole Number

Multiple Choice

The number is a/an...

• Rational Number

• Irrational Number

• Integer

• Whole Number

Multiple Choice

The number is a/an...

• Rational Number

• Irrational Number

• Integer

• Whole Number

Multiple Choice

The number is a/an...

• Rational Number

• Irrational Number

• Integer

• Whole Number

Multiple Choice

Which is an example of an integer?

• -9

• 0.567

• 1/2

• -0.957643...

Poll

Rate this math lesson on how helpful it was.

1 = not very helpful

3 = somewhat helpful

5 = very helpful

• 1

• 2

• 3

• 4

• 5

Complex Numbers

​

Multiple Choice

Simplify the following radical: √32

• 3√2
• 4√8
• 16√2
• 4√2

Multiple Choice

Simplify √-28

• -√28
• -2i√7
• 2i√7
• 28i

Multiple Choice

Simplify -√-48

• 4i√3
• -4i√3
• i√48
• -48i

Multiple Choice

Simplify √-23

• -√23
• 23i
• 23
• i√23

Multiple Choice

√-36

• 6
• -6
• -6i
• 6i

Multiple Choice

-3√80a⁷

• 4a³√5a
• -12a³√5a
• -3a³√5a
• -48a³√5a

Operations with complex numbers

Adding and Subtracting

Multiple Choice

Simplify: 
(10+ 15i) - (48 - 30i)

• 58 - 45i
• 58 - 15i
• -38 - 15i
• -38 + 45i

Multiple Choice

Find the sum.
(5-2i) + (-7+8i)

• -2+6i
• 12+6i
• -35-16i²
• -35 -16i

Multiple Choice

Simplify: 
(10+ 15i) - (48 - 30i)

• 58 - 45i
• 58 - 15i
• -38 - 15i
• -38 + 45i

Multiple Choice

Simplify:
-3i(4 + 2i)

• 6 - 12i
• 6 + 12i
• -12i - 6i²
• -12i + 6i²

Multiple Choice

Simplify:
(2 - 3i)(5 + 4i)

• 22 - 7i
• 22 + 7i
• 22 - 7i²
• 22 + 7i²

Multiple Choice

Simplify
(-6 - 2i)²

• 32
• 36-4i
• 36+24i
• 32+24i

Multiple Choice

(4-5i)(4+5i)

• 41
• -9
• 41+40i
• -9+40i

Imaginary & Complex Numbers

Alg 2 Lesson 3.3 & 3.4

What is an imaginary number?

Multiple Choice

 $\sqrt{-48}$  

•  $4i\sqrt{3}$  

•  $3i\sqrt{12}$  

•  $-4\sqrt{3}$  

•  $-8\sqrt{6}$  

Multiple Choice

 $\sqrt{-80}$  

•  $2i\sqrt{20}$  

•  $4i\sqrt{5}$  

•  $-8\sqrt{10}$  

Multiple Choice

 $3\sqrt{-90}$  

•  $3i\sqrt{10}$  

•  $9i\sqrt{10}$  

•  $27i\sqrt{10}$  

Complex Numbers

Real + Imaginary Number

a + bi

example: 2 + 3i

Multiple Choice

Add:

 $\left(-6\ +\ 10i\right)\ +\ \left(8\ +\ 4i\right)$  

•  $14+14i$  

•  $2+14i$  

•  $16i$  

Multiple Choice

Subtract:

 $\left(8+7i\right)-\left(6+3i\right)$  

•  $2+4i$  

•  $2+10i$  

•  $14+10i$  

•  $14+2i$  

Multiply Complex Numbers

Multiple Choice

 $5i\times10i$  

•  $50i^2$  

•  $-50$  

•  $50$  

•  $50i$  

Multiple Choice

 $\left(2i\right)\left(-9i\right)$  

•  $18$  

•  $-18$  

•  $-18i^2$  

•  $18i$  

Multiple Choice

 $\left(2+3i\right)\left(3+7i\right)$  
*Distribute

•  $6+21i^2$  

•  $-15+23i$  

Dividing Complex numbers

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Review of multiplying complex numbers

Remember i² = -1

Multiple Choice

(i)(3i)

• -3i
• -3
• 3
• 2i

Multiple Choice

(-3-i)(6-i)

• -21-7i
• -19-3i
• -15-5i
• -17-9i

Multiple Choice

(7-5i)(7+5i)

• 74
• 75
• 74-1

Find the conjugate

The conjugate of a + bi is a - bi

The conjugate of a - bi is a + bi

Multiple Choice

What is the conjugate of 2 + 3i?

• 2 - 3i

• -2 + 3i

• -2 - 3i

• 2 + 3i

Multiple Choice

What is the conjugate of -5 + 4i

• 5 + 4i

• 5 - 4i

• -5 - 4i

• -5 + 4i

Multiple Choice

What is the conjugate of 4i?

• 4i

• 4 + 4i

• 4-4i

• -4i

To divide complex numbers

Multiply the top and bottom by the conjugate of the denominator

Multiple Choice

 $\frac{5-8i}{3+2i}$  , what do you do to divide?

• Multiply the top and bottom by 3-2i

• Multiply the top and bottom by 5 + 8i

• multiply the top by 5 + 8i and the bottom by 3 - 2i

• Multiply the top and bottom by -3 - 2i

Multiple Choice

Simplify

Multiple Choice

• A
• B
• C
• D

Multiple Choice

• A
• B
• C
• D

Multiple Choice

• A
• B
• C
• D

Multiple Choice

Let $$f\left(x\right)=x+4$$ and $$g\left(x\right)=x^2+1$$ . Find $$f\left(-5\right)$$ and $$g\left(-5\right)$$

• f(-5) = 1

  g(-5) = -24

• f(-5) = -1

  g(-5) = 26

• f(-5) = -1

  g(-5) = 4

• f(-5) = 9

  g(-5) = 4

Multiple Choice

Let $$f\left(x\right)=x+4$$ and $$g\left(x\right)=x^2+1$$ . Find $$f\left(-5\right)+g\left(-5\right)$$

• -23

• 25

• 3

• 13

Multiple Choice

Let $$f\left(x\right)=x^2-x$$ and $$g\left(x\right)=x+2$$ . Find $$\left(g-f\right)\left(x\right)\ and\ \left(g-f\right)\left(-1\right)$$

• $$\left(g-f\right)\left(x\right)=-x^2+2x+2$$

  $$\left(g-f\right)\left(-1\right)=4$$

• $$\left(g-f\right)\left(x\right)=x^2-x-2$$

  $$\left(g-f\right)\left(-1\right)=-1$$

• $$\left(g-f\right)\left(x\right)=-x^2+2x+2$$

  $$\left(g-f\right)\left(-1\right)=-1$$

• $$\left(g-f\right)\left(x\right)=-x^2+2x+2$$

  $$\left(g-f\right)\left(-1\right)=0$$

Multiple Choice

• $$\left\{x\text{|}x\ne8\right\}$$

• $$\left\{x\text{|}x\ne\pm\sqrt[]{7},\ 8\right\}$$

• $$ℝ$$

Multiple Choice

Given $$f\left(x\right)=\frac{3}{2x+1}$$ and $$g\left(x\right)=x-4$$ . Find the domains of $$f+g,\ f-g,\ \text{and }f\cdot g$$

• $$ℝ$$

• $$\left\{x\text{|}x\ne-0.5,\ -4\right\}$$

• $$\left\{x\text{|}x\ne4\right\}$$

• $$\left\{x\text{|}x\ne-0.5\right\}$$

Multiple Choice

Given $$f\left(x\right)=\frac{3}{2x+1}$$ and $$g\left(x\right)=x-4$$ . Find the domains of $$f\text{/}g$$

• $$ℝ$$

• $$\left\{x\text{|}x\ne-0.5,\ 4\right\}$$

• $$\left\{x\text{|}x\ne-4\right\}$$

• $$\left\{x\text{|}x\ne-0.5\right\}$$

Homework