Download Math 96A Test 1

Math 96A Test 1
Flash Cards
Math 96 Test 1
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Real numbers & properties
Solve equations & inequalities
Absolute Value equations & inequalities
Translation word problems
Exponent Rules
Graph linear functions
Find equation of a line
Classify the given numbers.
Classify the given numbers.
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Natural: 1, 2, 3, 4, …
also will be whole, integer, rational and real
Whole: 0, 1, 2, 3, …
also will be integer, rational and real
Integer: … , -2, -1, 0, 1, 2, …
also will be rational and real
Rational: can be written as a fraction – decimals
with repeating or terminating decimals
Irrational: decimals with no repeating patterns
and they go forever
Real: all the above numbers are real numbers –
so far everything you know is a real number!
Classify the given numbers.
-2
Natural Numbers
Whole Numbers
Integers
Rational Number
Irrational Numbers
Real Numbers
0
11
.4545…
Classify the given numbers.
-2
0
11
.4545…
Natural Numbers
X
Whole Numbers
Integers
X
X
Rational Number
X
X
X
Irrational Numbers
Real Numbers
X
X
X
X
X
Name the properties of
Real numbers.
Name the properties of
Real numbers.
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Associative: something new inside the
parentheses – add and multiply
Commutative: something has moved its
location – add and multiply
Distributive: multiply on the outside, adding
in the inside
Identities: “it” will not change – add by zero
OR multiply by 1
Inverses: will make “it” go away – add the
opposite OR multiply by the reciprocal
Name the properties of
Real numbers.
Name the properties of
Real numbers.
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Multiplication Property of ZERO: if you
multiply BY zero you get zero!
Multiplication Property of ZERO: a (0) = 0
Closure: you get an answer! a + b = c
Trichotomy Property: 1 of 3 things must
be true a < b or a = b or a > b
Transitive Property: if a < b and b < c
then a < c
Name the properties of
Real numbers.
4•0 =0
Name the properties of
Real numbers.
4•0 =0
The Zero Product Property
Solve each equation for x.
Solve each equation for x.
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Step 1. identify the variable you are solving
for and clear parentheses
Step 2. clear fractions (multiply by the LCM)
and/or clear decimals (multiply by 10s)
Step 3. get just 1 variable
Step 4. get the variable alone, furthest first –
according to the reverse Order of Operations
Solve each equation for x.
5[2 – (2x – 4)] = 2(5 – 3x)
Solve each equation for x.
5[2 – (2x – 4)] = 2(5 – 3x)
5[2 – 2x + 4] = 2(5 – 3x)
5[– 2x + 6] = 2(5 – 3x)
-10x + 30 = 10 – 6x
-4x + 30 = 10
-4x = -20
x = 5
Solve each equation for x.
2 x 1 1 x  3
 
1
8
4
2
Solve each equation for x.
2 x 1 1 x  3
 
1
8
4
2
8  2 x  1 8 1 8  x  3


 8 1
8
4
2
2 x  1  2  4  x  3  8
2 x  1  4 x  12  8
2 x  3
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x  3/ 2
Graph the following Inequalities
Graph the following Inequalities
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Greater than and Less than – open circle
Greater than or equal to and Less than or
equal to – closed circle
If x comes first – go the same way as the
inequality
Space numbers evenly on the number
line, one variable – one line
Graph the following inequality
x > -2
Graph the following inequality
x > -2
-4
-2
0
2
4
6
Solve Inequalities for x,
and graph your solution.
Solve Inequalities for x,
and graph your solution.
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IF you multiply (or divide) by a
negative, the inequality will change
direction.
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Follow the rules for graphing
inequalities.
Solve this inequality for x,
and graph your solution
4 – 7x > -10
Solve this inequality for x,
and graph your solution
4 – 7x > -10
-7x > -14
(-1/7)(-7x) < (-1/7)(-14)
x<2
-4
-2
0
2
4
6
Multiplied by
a Negative
Solve for the indicated variable.
Solve for the indicated variable.
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Step 1. identify the variable you are solving
for and clear parentheses
Step 2. clear fractions (multiply by the LCM)
and/or clear decimals (multiply by 10s)
Step 3. get just 1 variable, factor if needed
Step 4. get the variable alone, furthest first –
according to the reverse order of operations
Solve for the indicated variable.
W = ab + ah;
solve for a
Solve for the indicated variable.
W = ab + ah; solve for a
Too many a’s – factor!
W = a (b + h)
W = a(b + h)
(b + h) (b + h)
W
=a
b+h
Solve the following equations
containing Absolute Value bars.
Solve the following equations
containing Absolute Value bars.
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Make sure you FIRST isolate the
absolute value bars
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2 Bars – 2 Problems – what can go into
the bars and come out as desired?
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Special case: | x | = negative
No Solution
Solve the following equation
containing Absolute Value bars.
| 2x – 1 | + 5 = 8
Solve the following equation
containing Absolute Value bars.
| 2x – 1 | + 5 = 8
| 2x – 1 | = 3
2x – 1 = 3
2x = 4
x = 2
2x – 1 = -3
2x = -2
x = -1
Solve each of the Absolute Value
Inequalities and graph.
Solve each of the Absolute Value
Inequalities and graph.
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Make sure you FIRST isolate the
absolute value bars
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2 Bars – 2 Problems – what can go
into the bars and come out as desired?
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Special cases:
| x | < negative
No Solution
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| x | > negative
all real numbers
Solve the Absolute Value
Inequality and graph.
| 4 – 2x | + 5 > 3
Solve the Absolute Value
Inequality and graph.
| 4 – 2x | + 5 > 3
| 4 – 2x | > -2
Always True,
Absolute Value is greater than a Negative
Translate these words and write
an equation then solve it.
Translate these words and write
an equation then solve it.
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Read the whole problem all the way through
at least once.
Write what you read as you read it
Sum – (add inside parentheses )
Total – (add inside parentheses )
Difference – (subtract inside parentheses)
Less than – write subtraction “backwards”
Subtracted from – write subtraction
backwards
Translate these words and write
an equation then solve it.
Five times the difference between three
and twice a number is negative five.
Translate these words and write
an equation then solve it.
Five times the difference between three
and twice a number is negative five.
5(3 – 2n) = -5
15 – 10n = -5
-10n = -20
n =2
The number is two!
End in Words!
Simplify the given expression.
Do not leave negative exponents.
Simplify the given expression.
Do not leave negative exponents.
a a  a
m
n
m n
m
a
m n
a
n
a
0
a 1
a 
m n
a
mn
a
m
1
 m
a
a negative exponent
"means" take the
reciprocal of the base
Simplify the given expression.
Do not leave negative exponents.
2
5
Simplify the given expression.
Do not leave negative exponents.
2
5
1
2
5
1
25
Simplify the given expression.
Do not leave negative exponents.
Simplify the given expression.
Do not leave negative exponents.
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Clear outside exponents first,
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move the “location” of the base that
has a negative exponent
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the base still has an exponent, but now
it is positive.
Simplify the given expression.
Do not leave negative exponents.
y y 
 5 2 
 y y 
2
1
2
Simplify the given expression.
Do not leave negative exponents.
y y 
 5 2 
 y y 
2
1
2
 y y 
 2 1 
y y 
5
2
2
10
4
y y
4 2
y y
14
y
12
y
2
y
Simplify the given expression.
Do not leave negative exponents.
Simplify the given expression.
Do not leave negative exponents.
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Clear outside exponents first, make
sure all parenthesis are “gone” before
“moving” bases.
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The base is only what the exponent
touches.
Simplify the given expression.
Do not leave negative exponents.
2
x y
4 3
x y
Simplify the given expression.
Do not leave negative exponents.
2
x y
4 3
x y
2
x yy
4
x
3
2
x y
4
x
4
4
y
2
x
Graph by Plotting Points
Graph by Plotting Points
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Use my favorite numbers -2, -1, 0, 1, 2
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Replace x with the value you have in
the table and find the value of y.
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(x, y) a point is an ordered pair of
numbers
First number, go along the x-axis
Second number, go in the y-axis
direction
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Graph by Plotting Points
y=½x–5
Graph by Plotting Points
y=½x–5
(-2, )
y = ½ (-2) – 5
y = -6
(-2, -6)
(4, )
y = ½ (4) – 5
y = -3
(4, -3)
Graph by Intercepts
Graph by Intercepts
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Let x = 0 to find the y-intercept,
the point on the y-axis.
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Let y = 0 to find the x-intercept,
the point on the x-axis.
Graph by Intercepts
2x – 4y = -8
Graph by Intercepts
2x – 4y = -8
2(0) – 4y = -8
-4y = -8
y = 2
(0, 2)
2x – 4(0) = -8
2x = -8
x = -4 (-4, 0)
Graph by using Slope-Intercept
form
Graph by using Slope-Intercept
form
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Solve for y:
y = mx + b
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b = y-intercept, start on y-axis
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from the “starting” point,
go up and over
rise
m
run
Graph by using Slope-Intercept
form
3x – 2y = 4
Graph by using Slope-Intercept
form
3x – 2y = 4
-3x
-3x
-2y = -3x + 4
(-½)(-2y) = (-½)(-3x + 4)
y = 3/2 x – 2
Graph the corresponding line on
the Cartesian coordinate system.
Graph the corresponding line on
the Cartesian coordinate system.
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Plot points, using an x-y table
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Graph using intercepts, two separate
points
(x, 0) and (0, y)
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Solve for y, graph using the slopeintercept form. Start on the y-axis, go
up/down and then over.
Graph by using any method
y = -2x + 3
Graph by using any method
y = -2x + 3
Find an equation for the line that
satisfies the given conditions.
Find an equation for the line that
satisfies the given conditions.
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Equation of a line: y = mx + b
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Point (x, y)
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Given two points stack & subtract to
find slope m
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in y = mx + b, replace x, y, and m to find b
Find an equation for the line that
satisfies the given conditions.
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Find the equation of the line containing the
two points (-3, 4) and (2, 1)
Find an equation for the line that
satisfies the given conditions.
Find the equation of the line containing the
two points (-3, 4) and (2, 1)
1 4
3
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y – y1 = m(x – x1)
2  (3) 5
 y – 1 = -3/5(x – 2)
clear the parentheses
 y – 1 = -3/5x + 6/5
clear the fraction,
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multiply by 5
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5(y – 1) = -3x + 6 Simplify
5y – 5 = -3x + 6 get all the variables on 1 side
3x + 5y = 11
and the constants on the other side
Find an equation for the line that
satisfies the given conditions.
Find an equation for the line that
satisfies the given conditions.
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Parallel lines have the same slope
Perpendicular lines have
opposite & reciprocal slope
Given an equation: Ax + By = C
Solve for y to find m
parallel use m
perpendicular use - 1/m
use the given point (x, y)
y = mx + b: replace x, y, and m to find b
Find an equation for the line that
satisfies the given conditions.
Perpendicular to 3x – y = 4 and passes
through the point (-3,6).
Find an equation for the line that
satisfies the given conditions.
Perpendicular to 3x – y = 4 and passes
through the point (-3,6).
Solve the equation for y – find the slope
-y = -3x + 4
y = 3x – 4
m = 3 use for perpendicular line m = -1/3
6 = (-1/3)(-3) + b
5=b
Equation: y = -1/3 x + 5