Algebra II, Chapter 1 Definitions
Algebra II Chapter 1: Solving Equations and Inequalities
Section 1.1: Expressions and Formulas
The graphing calculator does know the order of operations, but be careful with parenthesis.
Order of Operations: use the mnemonic PEMDAS:
Please (parentheses)
Excuse (exponents)
My (multiplication)
Dear (division) (multiply and divide in order from left to right)
Aunt (addition)
Sally (subtraction) (+ and - in order from left to right)
Variables: are symbols, usually letters (x, y, z) used to represent unknown quantities.
Algebraic expressions: expressions that contain at least one variable, you can also evaluate an algebraic expression by replacing each variable with a number and applying the order of operations (PEMDAS).
Formula: a mathematical sentence that expresses the relationship between certain quantities
Section 1.2: Properties of Real Numbers
Real Numbers: numbers commonly used everyday are real numbers, each real number corresponds to exactly one point on the number line, and every point on the number line represents exactly one real number. Real numbers can be classified as rational or irrational numbers.
1. Irrational Numbers: (I) a real number that is not rational is irrational. The decimal form of an irrational number neither terminates nor repeats. Any non perfect roots.
Ex.[pic][pic], 0.0100010001…
2. Rational Numbers: (Q) a rational number can be expressed as a ratio, [pic] , where m and n are integers and n is not equal to zero. The decimal form of a rational number is either a terminating or repeating decimal: Ex. 1.9,[pic], [pic], 0, all fractions
Integers: (Z) positive and negative whole numbers, including zero (whole numbers and their opposites): Ex. … -2, -1, 0, 1, 2 …
Whole: (W) counting numbers including zero: Ex. 0, 1, 2, 3 …
Naturals: (N) counting numbers (doesn’t include zero): Ex. 1, 2, 3 …
Properties of Real Numbers:
|Property |Addition |Multiplication |
|Commutative |a + b = b + a |ab = ba |
|Associative |(a + b) + c = a + (b + c) |(ab)c = a(bc) |
|Identity |a + 0 = a = 0 + a |a(1) = a = (1)a |
|Inverse |a + (-a) = 0 = (-a) + a |If a does not equal 0, a ([pic]) = 1 = ([pic])a |
|Distributive |a(b + c) = ab + ac and (b + c)a |= ba + ca |
Commutative Property: Commuting distance is the same from work to home as it is from home to work, you communicate your order at the Sonic, therefore commutative is to change the order of the operation with the CP: a + b = b + a
Associative Property: You associate with your group of friends, the AP is a “group” of friends: a + (b + c) = (a + b) + c
Distributive Property: To distribute a paper is to give it to everyone.
The DP, “gives” whatever is outside the parentheses to everything inside: a(b + c) = ab + ac
Identity: The identity for addition and multiplication is the number you can add or multiply with to get the identical answer.
Section 1.3: Solving Equations
Open Sentence: a mathematical sentence containing one or more variables.
Equation: mathematical sentence stating that two mathematical expressions are equal.
Section 1.4: Solving Absolute Value Equations
Absolute Value: [pic], of a number is its distance from 0 on the number line. Since distance is nonnegative, the absolute value of a number is ALWAYS positive.
Empty Set: no solution to an equation. Absolute value cannot be equal to a negative number.
Section 1.5: Solving Inequalities
[pic] Less than: Shade LEFT
[pic]Greater than: Shade RIGHT
FLIP THE SYMBOL: MULTIPLY OR DIVIDE BY A NEGATIVE NUMBER
Section 1.6: Solving Compound and Absolute Value Inequalities
Compound Inequality: consists of two inequalities joined by the word and or the word or. You must solve each part of the inequality.
Intersection: the graph of a compound inequality containing and.
Union: the graph of a compound inequality containing or.
AND: INTERSECTION OR: DOES NOT INTERSECT
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