Chapter 1: Equations and Inequalities

1-1: Expressions and Formulas

variable: symbols, usually letters, that represent unknowns (i.e. x, y, or z)

algebraic expression: an expression that contains at least one variable (i.e. x + 3)

order of operations: used to solve expressions and equations, used to solve for variables

Step 1: evaluate expressions inside grouping symbols

Step 2: evaluate all powers

Step 3: multiply and/or divide from left to right

Step 4: add and/or subtract from left to right

monomial: an algebraic expression that is composed of a number, a variable, or the product of a number and a variable (i.e. 5b, -x, 6xy)

• these cannot contain variables in the denominator, variables with negative exponents, or variables inside radicals

constants: monomials that contain no variables (i.e. 23, 5, -2)

coefficient: the numerical part of a monomial with a variable (i.e. 5x, 5 is the coefficient)

degree: the sum of the exponents of the variables in a monomial (i.e. x^2y^2, the degree is 4)

• the degree of a constant is 0

power: an expression with the form x^n - also used to refer to the exponent itself

polynomial: a monomial OR a sum of monomials

terms: the monomials that make up a polynomial (i.e. 3x^2 + 5x, the terms are 3x^2 and 5x)

like terms: terms that can be combined (i.e. 2x + 4x, these are like terms because they can sum to 6x)

trinomial: three unlike terms (i.e. x^2 + 3x - 1, none of the terms can be combined)

binomial: two unlike terms (i.e. x + y)

formula: a mathematical sentence that expresses the relationship between certain quantities - can be used to solve for certain values, if others are given

• formula for the area (A) of a trapezoid: A = 1/2 h(b1 + b2), where h is the height, and b1 and b2 are the lengths of the bases
  • if three out of four of these variables are given, one is able to use the formula to find the remaining variable

1-2: Properties of Real Numbers

real numbers: numbers that correspond to exactly one point on the number line, every point represents

rational numbers: a number that can be expressed as a ratio, where the numerator and denominator are integers, and the denominator is NOT zero

• the decimal form is either a repeating, or terminal

irrational numbers: a number that is NOT rational, the decimal form doesn’t terminate NOR repeat

Properties of Real Numbers

For any real numbers a, b, and c:

Distributive: a(b + c) = ab + ac AND (b + c)a = ba + ca

Simplifying Expressions

2(5m + n) + 3(2m - 4n)

= 2(5m) + 2(n) + 3(2m) - 3(4n)

= 10m + 2n + 6m - 12n

= 10 + 6m + 2n - 12n

= (10 + 6)m + (2 - 12)n

= 16m - 10n

1-3: Solving Equations

open sentence: a mathematical sentence with one or more variables

equation: a mathematical sentence that sets two expressions equal to each other

solution: whenever a variable is replaced by a number

Properties of Equality

Addition and Subtraction

symbols: for any real numbers a, b, and c, if a = b, then a + c = b + c, AND a - c = b - c

examples:

1. if x - 4 = 5, then x - 4 + 4 = 5 + 4
2. if n + 3 = -11, then n + 3 - 3 = -11 - 3

Multiplication and Division

symbols: for any real numbers a, b, and c, if a = b, then a • c = b • c, AND if c ≠ 0, a/c = b/c

examples:

1. if m/4 = 6, then 4 • m/4 = 4 • 6
2. if -3y = 6, then -3y/-3 = 6/-3

1-4: Solving Absolute Value Equations

absolute value: the distance of a number from 0 on a number line, represented by |x|

• for any real number a, if a is positive or zero, the absolute value of a is a; if a is negative, the absolute value of a is the opposite of a

empty set: solution set of an equation with no solution set

1-5: Solving Inequalities

Trichotomy Property: for any two real numbers, a and b, exactly ONE of the following statements is true:

1. a < b
2. a = b
3. a > b

set-builder notation: a way to express a solution set of an inequality

• { x | x > 9} → the set of all numbers x such that x is greater than 9

Properties of Inequality

These properties also hold true for ≤, ≥, and ≠

Addition Property

for any real numbers a, b, and c:

1. if a > b, then a + c > b + c
2. if a < b, then a + c < b + c

Subtraction Property

for any real numbers a, b, and c:

1. if a > b, then a - c > b - c
2. if a < b, then a - c < b - c

Multiplication Property

for any real numbers a, b, and c, where

c is positive:

1. if a > b, then ac > bc
2. if a < b, then ac < bc

c is negative:

1. if a > b, then ac < bc
2. if a < b, then ac > bc

Division Property

for any real numbers a, b, and c, where

c is positive:

1. if a > b, then a/c > b/c
2. if a < b, then a/c < b/c

c is negative:

1. if a > b, then a/c < b/c
2. if a < b, then a/c > b/c

1-6: Solving Compound and Absolute Value Inequalities

compound inequality: two inequalities joined by the word and or the word or

intersection: the set of elements common to two sets, the graph of a compound inequality containing and

union: the set of elements belonging to one or more of a group of sets, the graph of a compound inequality containing or

Absolute Value Inequalities

for all real numbers a and b, b > 0, the following statements are true

1. if |a| < b, then -b < a < b
2. if |a| > b, a > b OR a < -b