Classwork #40: Final Review: Solving Equations, Word Problems, Linear Equations, Systems of

Linear Equations

Solving Equations: Isolate the variable!

Things to watch for:

____________________ when you have parentheses

_________ the inequality sign when ____ or ____ by a negative

0 = 0 means _________________

0 = 3 means _________________

Solve. If it is an inequality, include a graph.

1.) 3 5

6 4w w 2.) 24 – 2(2m + 1) = -6 3.) 3 2 1 13x

4.) 4v + 3 < -1 or -2v + 7 < 1 5.) 6m – 5 = 7m + 7 – m 6.) 10 – 8a = 2(5 – 4a)

Solving Word Problems Use a chart!

Distance Problems

D = R x T

Same direction ____

Opposite direction ____

Round trip _____

Mixture Problems

Volume x %/$ = amount

Volumes: x and (total) – x

7.) Two cars leave town at the same time heading in the same direction. One car travels 60mph and

the other travels at 40mph. After how many hours will they be 50 miles apart?

Distance Rate Time

Slow Car

Fast Car

8.) Two cars leave town at the same time heading in the opposite direction. One car travels 60mph

and the other travels at 30mph. After how many hours will they be 150 miles apart?

Distance Rate Time

Slow Car

Fast Car

9.) Raisins cost $2 per pound and nuts cost $5 per pound. How many pounds of each should you use

to make a 30-lb mixture that costs $4 per pound?

Raisins

Nuts

Mixture

Equations of Lines:

Start with slope: m = ________

Parallel means _________ slope

Perpendicular means ______, _______ slope

Plug in m and point (x, y) into

_____________

Solve for b

Plug m and b into ___________

Covert to standard form, if necessary

10.) Find the slope of the line passing through

the pair of points:

a) (-4, 7) and (2, 9)

b) (3, 8) and (3, -1)

Slope-Intercept Form Standard Form Intercepts

11.) Write the equation in slope-intercept form.

State the slope (m) and y-intercept (b)

12

3x y

12.) Write the equation in standard form and find

the x and y intercepts.

36

4y x

13.) Write the equation of the line with slope of 2

7 and y-intercept 4 in standard form.

14.) Graph the line by finding the slope and y-

intercept. 4 2 6x y

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10

x

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

9

10

y

15.) Graph the line by finding the x and y

intercepts.

2 3x y

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10

x

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

9

10

y

16.) Write the equation of the line with slope -4

and passing through the point (6, -1) in standard

form.

17.) Write the equation of the line passing through

the points (-3, 2) and (1, 4) in slope-intercept form

18.) Write the equation of the line parallel to

y = -2x + 3 and passing through the point (4, -1) in

standard form.

19.) Write the equation of the line perpendicular

to 3x – y = 2 and passing through the point (-1, -2)

in standard form.

21.) Solve for a and b using the substitution

method

2 4

3 4 15

a b

a b

22.) Solve for x and y by using the

addition/elimination method:

2 5 3

3 6 9

x y

x y

Classwork #41: Rules of Exponents

When you are…

Adding/Subtracting

terms

- Only combine

the like terms

- Leave variables

alone

- Only

add/subtract

coefficients

Ex: 3x2 – 5y + x

2 + y

Multiplying Terms

- Multiplying

coefficients together

- Add exponents on

the like variable

expressions

Ex: (2a3b

2)(-4a

2b

5)

Dividing Terms

- Divide/reduce

coefficients

- Subtract

exponents on like

variables

Ex: 6 12

3 18

18

12

s t

s t

Raising Terms to a

Power

- Raise the

coefficient to

the power

- Multiply the

exponents

Ex: (-3w4z

3)2

Watch out for negative exponents: Ex: 4p-3

Ex: (9w3y

-4)(-3w

-1y

2) Ex:

2 5

2

12

3

x y

xy

Simplify: Leave all answers in positive exponents

1.) (3bc8)(-5b

2c)(2b

3c

-2) 2.) (-5m

-2n

3)3 3.) (-3x

2y

3)2(-8xy

5)

4.) 7 5

9

27

3

m n

m n 8.)

22 2

3

4 9

3 2

xy xz

z y 9.)

21

3

3

4

t

r

Absolute Value: Isolate the | |, and write 2 equations!

Things to watch for:

- In an inequality, you must:

- get the | | on the left side

- if it is < or , it is a ________ _____ (a sandwich)

- if it is > or , it is a ________ _____ (a sandwich)

Solve each absolute value equality:

1.) 2 5 11p 2.) 3 5 6r

Solve and graph each absolute value inequality:

3.) 5 14d 4.) 3 8k

Graphing Linear Inequalities

- Get y alone

- Graph using slope and y-intercept

- Line: > or < means _______________

or means ______________

- Shading: < or means

_______________

> or means

_______________

5.) Graph the linear inequality: 5 2 9x y

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10

x

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

9

10

y

6.) Graph the set of linear inequalities and shade

the solution region.

2 1

4 3

y x

x y

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10

x

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

9

10

y

Square Roots:

- Adding/Subtracting: You can only add

and subtract terms with like radicals

5 3 2 2 5 3 2 3

- Multiplying: multiply outside numbers

together, and inside numbers together.

Then reduce.

6 6 5 10

- Dividing: you can’t have a radical on

the

10

3 2

6

5 3

Simplify:

7. 23 2 2 10g g h h gh 8. 2

7 5 9. 48

81

b 10.

5 2

6m

11.) 2 28 6 63 12.) 5 2 7 2 6 13.) 4 3 2 2 5 3 2

14.) 5 15 3 2

3 15.)

6

3 2 16.) Solve for b:

b15

9

For 17-18, solve and check your solution(s):

17.) 4 2 5x 18.) 3 2 4x x

Classwork #42: Scientific Notation

Converting from Standard Notation to Scientific Notation

- Place the decimal point so that you have 1 digit to the left of the decimal point.

____.____ ____ ____

- Count the number of places you need to move the decimal to get back to your

original number. This number will be the exponent on the 10.

- If you are moving to the left, this exponent will be ________________

- If you are moving to the right, this exponent will be _______________

Write in scientific notation:

1.) 13,000 2.) 0.00063 3.) -0.05 4.) 125.7 x 108 5.) 0.045 x 10

-7

Multiplying Numbers in Scientific Notation

- Multiply coefficients (decimal terms)

- Add the exponents

- Check that your answer is still in scientific notation

6.) (2.5 x 105)(3.0 x 10

8) 7.) (5.4 x 10

-2)(6.2 x 10

-6) 8.) (9.3 x 10

-4)(3.1 x 10

5)

Dividing Numbers in Scientific Notation

- Divide coefficients (decimal terms)

- Subtract the exponents

- Check that your answer is still in scientific notation

9.) 8

2

4.8 x 10

2.4 x 10 10.)

5

2

4.8 x 10

6.0 x 10 11.)

10

4

1.2 x 10

3.0 x 10

Taking Numbers in Scientific Notation to a Power

- Raise the coefficient to the given power

- Multiply the exponents

- Check that your answer is still in scientific notation

12.) (1.2 x 104)2

13.) (2 x 10-9

)-2

14.) (-3 x 105)3 15.) (8 x 10

-5)2

Solve by factoring

16.) 12k2 – 5k = 2 17.) 49m

2 – 16 = 0

Solve by using square roots

18.) 4w2 = 18

19.) 3y2 – 8 = 0

20.) 5m2 – 16 = 0 21.) (2x – 10)

2 = 20

Solve by using the quadratic formula: Write down the formula:

22.) x2 – 10x + 7 = 0 23.) 4m

2 +12m = 7

Without graphing, find the information for each parabola:

24.) 2( ) 2 8 3f x x x

Vertex: ________; Axis of Symmetry: _________

What is the direction of opening? _______

Is the vertex a max or min? _______

Wider or narrower than y = x2

? ___________

25.) 23( ) 6 5

2f x x x

Vertex: ________; Axis of Symmetry: _________

What is the direction of opening? _______

Is the vertex a max or min? _______

Wider or narrower than y = x2 ? ___________

Simplify:

26.) 2

2

18 2

2 2 24

x

x x 27.)

2 2

2 2

6 13 6 6 2

4 9 4 1

x x x x

x x

28.) Divide using long division: 29.) Add:

4 24 2 16 1x x x 3 4

2 6 6 18x x

30.) Subtract: 31.) Solve:

2

2

1 1

x

x x

3

2 1

x x

x x

32.) Solve:

2

4 151

5 25t t

33.) John can weed the garden in 4hours. His wife can weed the same garden in 3 hours. How long

would it take them to weed if they worked together?

John

John’s Wife