CST Algebra 1 Study Guide – Flashcards
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rational number
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any number that can be expressed as the ratio of two integers in the form a/b where b≠0
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irrational number
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a real number that cannot be written in the form a/b where a and b are integers
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integer
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any of the natural numbers (positive or negative) or zero
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associative property
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addition: a+(b+c)=(a+b)+c multiplication: a(b×c)=(a×b)c
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commutative property
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addition: a+b=b+a multiplication: a×b=b×a
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distributive property
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a(b+c)=a×b+a×c
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identity property
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addition: a+0=a multiplication:a×1=a
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multiplicative inverse
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(mathematics) one of a pair of numbers whose product is 1: the reciprocal of 2/3 is 3/2
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reciprocal
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multiplicative inverse
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solving inequalities
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-3x+5 > 17 -3x > 12 x < -4
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absolute value equations
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| x-5 |=3 x - 5 = 3 or x - 5 = -3
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disjunction
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| x-5 | > 3 x-5 > 3 or x-5 < -3
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conjunction
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| x-5 | < 3 -3 < x-5 < 3
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scientific notation
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1,230,000 = 1.23 × 10⁶ 0.000504 = 5.04 × 10⁻⁴
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multiply exponents
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multiplying powers with the same base by adding exponents 7³ × 7⁵ = 7⁸
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divide exponents
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dividing powers with the same base by subtracting exponents 7⁵ / 7³ = 7²
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negative exponents
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5⁻³ = 1 / 5³ (one over 5 cube)
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zero as an exponent
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8⁰ = 1
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powers of powers
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multiplying the two powers together (5³)⁴ = 5¹²
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powers of products
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multiplying all the numbers inside parenthesis with the power outside (4ab⁴)³ = 64a³b¹²
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factoring polynomials
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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
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slope intercept form
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y = mx + b m - slope (rise over run) b - y-intercept (where the line crosses the y-axis)
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point-slope equation
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y - y₁ = m (x - x₁)
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slope
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m = y₂ - y₁ / x₂ - x₁
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multiplying rational expressions
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(x² + x) / x² × (3x - 3) / (x² - 1) = [x(x+1) × 3(x - 1)] / x² (x+1)(x-1) = 3 / x
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subtracting rational expressions
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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)]
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system of equations: addition method
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5x − 3y = 10 2x + 3y = 4 (this line is added by logic) 7x = 14
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system of equation: substitution method
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3x + 2y = 4 x = y -5 3 (y-5) + 2y = 4
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work problems
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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)
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motion problems
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mixture problems
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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
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rationalizing the denominator
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√2 / √3 = (√2√3) / (√3√3) = √6 / 3
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cube roots
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³√1 = 1 ³√8 = 2 ³√27 = 3 125∧1/3 = 5 ax³ + bx² + cx +d = 0
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pythagorean theorem
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a² + b² = c²
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graphing quadratic equations
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axis of symmetry x= -b / 2a
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functions
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domain - the first coordinates of a relation range - the second coordinates of a relation
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completing the square
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2x² 12x - 8 = 5 x² +6x = 13 x² + 6x +9 = 22 (x+3)² = 22 x+3 = ±√22 x = -3 ±√22
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quadratic formula
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x = [ -b ±√b²-4ac ] / 2a
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discriminant
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b² 4ac positive - two solutions zero - one solution negative - no solution
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graphing linear inequalities
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y > 2x -1 y ≤ x + 3
