Math 96A Test 1
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Transcript Math 96A Test 1
Math 96A Test 1
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Math 96 Test 1
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.
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.
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.
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.
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
x 3/ 2
Graph the following Inequalities
Graph the following Inequalities
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.
IF you multiply (or divide) by a
negative, the inequality will change
direction.
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.
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.
Make sure you FIRST isolate the
absolute value bars
2 Bars – 2 Problems – what can go into
the bars and come out as desired?
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.
Make sure you FIRST isolate the
absolute value bars
2 Bars – 2 Problems – what can go
into the bars and come out as desired?
Special cases:
| x | < negative
No Solution
| 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.
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.
5
2
Simplify the given expression.
Do not leave negative exponents.
5
2
1
2
5
1
25
Simplify the given expression.
Do not leave negative exponents.
Simplify the given expression.
Do not leave negative exponents.
Clear outside exponents first,
move the “location” of the base that
has a negative exponent
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.
Clear outside exponents first, make
sure all parenthesis are “gone” before
“moving” bases.
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
Use my favorite numbers -2, -1, 0, 1, 2
Replace x with the value you have in
the table and find the value of y.
(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
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
Let x = 0 to find the y-intercept,
the point on the y-axis.
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
Solve for y:
y = mx + b
b = y-intercept, start on y-axis
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.
Plot points, using an x-y table
Graph using intercepts, two separate
points
(x, 0) and (0, y)
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.
Equation of a line: y = mx + b
Point (x, y)
Given two points stack & subtract to
find slope m
in y = mx + b, replace x, y, and m to find b
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)
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
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,
multiply by 5
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.
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