Math 96A Test 1 Flash Cards Math 96 Test 1 Real numbers & properties Solve equations & inequalities Absolute Value equations & inequalities Translation

Math 96A Test 1

Flash Cards

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.


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!


-2 0 .4545…

Natural Numbers

Whole Numbers

Integers

Rational Number

Irrational Numbers

Real Numbers

11


-2 0 .4545…

Natural Numbers

Whole Numbers X

Integers X X

Rational Number X X X

Irrational Numbers X

Real Numbers X X X X

11

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



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


4 • 0 = 0


4 • 0 = 0

The Zero Product Property

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


5[2 – (2x – 4)] = 2(5 – 3x)


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 1 1 3

18 4 2

x x

Solve each equation for x. 2 1 1 3

18 4 2

x x

8 2 1 8 1 8 38 1

8 4 2

x x

2 1 2 4 3 8x x

2 1 4 12 8x x

2 3x 3/ 2x

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

Multiplied by a Negative

-4 -2 0 2 4 6

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


W = ab + ah; solve for a


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 – 1 = -3

2x = 4

x = 2

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 | x | > negative No Solution 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.


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


Five times the difference between three

and twice a number is negative five.


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.


0 1

m n m n

mm n

n

nm mn

a a a

aa

a

a

a a

1

a negative exponent

"means" take the

reciprocal of the base

mm

aa


25


25

2

1

51

25



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.


22 1

5 2

y y

y y


22 1

5 2

y y

y y

25 2

2 1

y y

y y

10 4

4 2

y y

y y14

122

y

yy



Clear outside exponents first, make sure all parenthesis are “gone” before “moving” bases.

The base is only what the exponent touches.


2

4 3

x y

x y


2

4 3

x y

x y 2 3

4

x yy

x 2 4

4

x y

x 4

2

y

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 Pointsy = ½ x – 5

Graph by Plotting Pointsy = ½ x – 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 Intercepts2x – 4y = -8

Graph by Intercepts2x – 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


Solve for y: y = mx + b

b = y-intercept, start on y-axis

from the “starting” point, go up and over

risem

run


3x – 2y = 4


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 slope-intercept 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.


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 the equation of the line containing the two points (-3, 4) and (2, 1)


Find the equation of the line containing the two points (-3, 4) and (2, 1)

y – y1 = m(x – x1) 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

1 4 3

2 ( 3) 5



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


Perpendicular to 3x – y = 4 and passes through the point (-3,6).


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

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Math 96A Test 1 Flash Cards Math 96 Test 1 Real numbers & properties Solve equations & inequalities Absolute Value equations & inequalities Translation