40 terms

CST Algebra 1 Study Guide

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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
commutative property
addition: 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
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
absolute value equations
| x-5 |=3
x - 5 = 3 or x - 5 = -3
disjunction
| x-5 | > 3
x-5 > 3 or x-5 < -3
conjunction
| x-5 | < 3
-3 < x-5 < 3
scientific notation
1,230,000 = 1.23 × 10⁶
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
slope intercept form
y = mx + b
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
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)]
system of equations: addition method
5x − 3y = 10
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
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)
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
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
pythagorean theorem
a² + b² = c²
graphing quadratic equations
axis of symmetry
x= -b / 2a
functions
domain - the first 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
quadratic formula
x = [ -b ±√b²-4ac ] / 2a
discriminant
b² 4ac
positive - two solutions
zero - one solution
negative - no solution
graphing linear inequalities
y > 2x -1 y ≤ x + 3
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