Algebra 1 Regents Review Flashcards

Study Notes

Additive Inverse: The opposite of a number; e.g., -a/2 becomes a/2, and -5 becomes 5.
Multiplicative Inverse: The reciprocal of a number; e.g., 5 becomes 1/5, and 2/3 becomes 3/2.
Associative Property: Changes grouping of numbers; e.g., A(BC) changes to B(AC).
Commutative Property: Changes order of numbers; e.g., a + b = b + a and ab = ba.
Rational Numbers: Any number that can be expressed as a fraction.
Rational Expression: An algebraic fraction with polynomial numerator and denominator, where the denominator cannot be zero.

Fraction Undefined Conditions:
- 6/x is undefined when x = 0.
- x/(x+2) is undefined when x = -2.
- (x² - 9)/(x² + 7x + 10) is undefined for x = -2 and x = -5.

Steps to Simplify:
- Factor both numerator and denominator.
- Simplify the factors.
- Eliminate common factors between numerator and denominator.
- Example: 42/63 simplifies to 2/3.
Example Simplifications:
- 56x²y/70x³y² simplifies to 4/5xy.
- (x² - 2x - 15)/(x² - x - 12) simplifies to (x - 5)/(x - 4).
- Multiplication and division can also be simplified similarly.

Intersection of Sets: Elements common to multiple sets, denoted by ∩; e.g., A ∩ B = {1, 3, 5} for A = {1, 3, 5, 7, 9} and B = {1, 2, 3, 4, 5, 6}.
Union of Sets: All elements from both sets without duplicates, denoted by ∪; e.g., E ∪ F = {1, 2, 3, 4, 5, 6, 7, 8, 9} for E = {1, 3, 5, 7, 9} and F = {2, 4, 6, 8}.

Complement: The elements not in a specified set; e.g.:
- Complement of A = {1, 2, 3, 4, 5} for A = {6, 7}.
- Complement of B = {1, 2, 4, 6} for B = {3, 5, 7}.
- Complement of A ∩ Complement of B = {1, 2, 4}.

Circumference: Distance around a circle; C = 2πr or C = πd.
Area Formulas:
- Triangle: A = (1/2)bh.
- Rectangle: A = bh.
- Square: A = s².
- Trapezoid: A = 1/2(b₁ + b₂)h, e.g., A = 806 for bases 20 and 42, height 26.

Perimeter of Equilateral Triangle: For side length represented by x + 5, perimeter is 3(x + 5) = 3x + 15.
Perimeter of Rectangle: Length (x) is less than width (x - 3) can be solved for given perimeter conditions, resulting in:
- Length = 34 yards, Width = 31 yards.

Circumference Calculation Example: For a circle with a diameter of 22, C ≈ 69.12 rounded to nearest hundredth.
Area of Shaded Region: Found by determining the total area and subtracting the area of unshaded portions.