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