40 terms

rational number

any number that can be expressed as the ratio of two integers in the form a/b where b≠0

irrational number

a real number that cannot be written in the form a/b where a and b are integers

integer

any of the natural numbers (positive or negative) or zero

associative property

addition: a+(b+c)=(a+b)+c

multiplication: a(b×c)=(a×b)c

multiplication: a(b×c)=(a×b)c

commutative property

addition: a+b=b+a

multiplication: a×b=b×a

multiplication: a×b=b×a

distributive property

a(b+c)=a×b+a×c

identity property

addition: a+0=a

multiplication:a×1=a

multiplication:a×1=a

multiplicative inverse

(mathematics) one of a pair of numbers whose product is 1: the reciprocal of 2/3 is 3/2

reciprocal

multiplicative inverse

solving inequalities

-3x+5 > 17

-3x > 12

x < -4

-3x > 12

x < -4

absolute value equations

| x-5 |=3

x - 5 = 3 or x - 5 = -3

x - 5 = 3 or x - 5 = -3

disjunction

| x-5 | > 3

x-5 > 3 or x-5 < -3

x-5 > 3 or x-5 < -3

conjunction

| x-5 | < 3

-3 < x-5 < 3

-3 < x-5 < 3

scientific notation

1,230,000 = 1.23 × 10⁶

0.000504 = 5.04 × 10⁻⁴

0.000504 = 5.04 × 10⁻⁴

multiply exponents

multiplying powers with the same base by adding exponents 7³ × 7⁵ = 7⁸

divide exponents

dividing powers with the same base by subtracting exponents 7⁵ / 7³ = 7²

negative exponents

5⁻³ = 1 / 5³ (one over 5 cube)

zero as an exponent

8⁰ = 1

powers of powers

multiplying the two powers together (5³)⁴ = 5¹²

powers of products

multiplying all the numbers inside parenthesis with the power outside (4ab⁴)³ = 64a³b¹²

factoring polynomials

1 - Always look for a common factor.

2 - Then look at the number of terms.

2 Terms - Difference of Squares

3 Terms - Trinomial Square

1x² + bx + c

ax² + bx + c

4 Terms - Factor by Grouping

3 - Always factor completely

2 - Then look at the number of terms.

2 Terms - Difference of Squares

3 Terms - Trinomial Square

1x² + bx + c

ax² + bx + c

4 Terms - Factor by Grouping

3 - Always factor completely

slope intercept form

y = mx + b

m - slope (rise over run)

b - y-intercept (where the line crosses the y-axis)

m - slope (rise over run)

b - y-intercept (where the line crosses the y-axis)

point-slope equation

y - y₁ = m (x - x₁)

slope

m = y₂ - y₁ / x₂ - x₁

multiplying rational expressions

(x² + x) / x² × (3x - 3) / (x² - 1)

= [x(x+1) × 3(x - 1)] / x² (x+1)(x-1)

= 3 / x

= [x(x+1) × 3(x - 1)] / x² (x+1)(x-1)

= 3 / x

subtracting rational expressions

3 / (x+1) − 5 / (x-1)

= {[(x-1)3] / [(x-1)(x+1)]} − {[5(x+1)] / [(x-1)(x+1)]}

= [3x-3-(5x+5)] / [(x-1)(x+1)]

= (3x-3-5x-5) / [(x-1)(x+1)]

= [-2(x+4)] / [(x-1)(x+1)]

= {[(x-1)3] / [(x-1)(x+1)]} − {[5(x+1)] / [(x-1)(x+1)]}

= [3x-3-(5x+5)] / [(x-1)(x+1)]

= (3x-3-5x-5) / [(x-1)(x+1)]

= [-2(x+4)] / [(x-1)(x+1)]

system of equations: addition method

5x − 3y = 10

2x + 3y = 4 (this line is added by logic)

7x = 14

2x + 3y = 4 (this line is added by logic)

7x = 14

system of equation: substitution method

3x + 2y = 4

x = y -5

3 (y-5) + 2y = 4

x = y -5

3 (y-5) + 2y = 4

work problems

It takes painter A 3 hours. It takes painter B 5 hours. How long would it take them, working together?

1/3 +1/5 = 1/x or (a×b) / (a+b)

1/3 +1/5 = 1/x or (a×b) / (a+b)

motion problems

...

mixture problems

One solution is 80% acid and another one is 30% acid. How much of each solution is needed to make a 200L solution that is 62% acid?

Quantity: a + b = 200

Acid: (0.8)a + (0.3)b = (0.62)200

Quantity: a + b = 200

Acid: (0.8)a + (0.3)b = (0.62)200

rationalizing the denominator

√2 / √3 = (√2√3) / (√3√3) = √6 / 3

cube roots

³√1 = 1 ³√8 = 2 ³√27 = 3 125∧1/3 = 5

ax³ + bx² + cx +d = 0

ax³ + bx² + cx +d = 0

pythagorean theorem

a² + b² = c²

graphing quadratic equations

axis of symmetry

x= -b / 2a

x= -b / 2a

functions

domain - the first coordinates of a relation

range - the second coordinates of a relation

range - the second coordinates of a relation

completing the square

2x² 12x - 8 = 5

x² +6x = 13

x² + 6x +9 = 22

(x+3)² = 22

x+3 = ±√22

x = -3 ±√22

x² +6x = 13

x² + 6x +9 = 22

(x+3)² = 22

x+3 = ±√22

x = -3 ±√22

quadratic formula

x = [ -b ±√b²-4ac ] / 2a

discriminant

b² 4ac

positive - two solutions

zero - one solution

negative - no solution

positive - two solutions

zero - one solution

negative - no solution

graphing linear inequalities

y > 2x -1 y ≤ x + 3