Algebra Tutorials!  
Monday 15th of July
Rotating a Parabola
Multiplying Fractions
Finding Factors
Miscellaneous Equations
Mixed Numbers and Improper Fractions
Systems of Equations in Two Variables
Literal Numbers
Adding and Subtracting Polynomials
Subtracting Integers
Simplifying Complex Fractions
Decimals and Fractions
Multiplying Integers
Logarithmic Functions
Multiplying Monomials
The Square of a Binomial
Factoring Trinomials
The Pythagorean Theorem
Solving Radical Equations in One Variable
Multiplying Binomials Using the FOIL Method
Imaginary Numbers
Solving Quadratic Equations Using the Quadratic Formula
Solving Quadratic Equations
Order of Operations
Dividing Complex Numbers
The Appearance of a Polynomial Equation
Standard Form of a Line
Positive Integral Divisors
Dividing Fractions
Solving Linear Systems of Equations by Elimination
Multiplying and Dividing Square Roots
Functions and Graphs
Dividing Polynomials
Solving Rational Equations
Use of Parentheses or Brackets (The Distributive Law)
Multiplying and Dividing by Monomials
Solving Quadratic Equations by Graphing
Multiplying Decimals
Use of Parentheses or Brackets (The Distributive Law)
Simplifying Complex Fractions 1
Adding Fractions
Simplifying Complex Fractions
Solutions to Linear Equations in Two Variables
Quadratic Expressions Completing Squares
Dividing Radical Expressions
Rise and Run
Graphing Exponential Functions
Multiplying by a Monomial
The Cartesian Coordinate System
Writing the Terms of a Polynomial in Descending Order
Quadratic Expressions
Solving Inequalities
Solving Rational Inequalities with a Sign Graph
Solving Linear Equations
Solving an Equation with Two Radical Terms
Simplifying Rational Expressions
Intercepts of a Line
Completing the Square
Order of Operations
Factoring Trinomials
Solving Linear Equations
Solving Multi-Step Inequalities
Solving Quadratic Equations Graphically and Algebraically
Collecting Like Terms
Solving Equations with Radicals and Exponents
Percent of Change
Powers of ten (Scientific Notation)
Comparing Integers on a Number Line
Solving Systems of Equations Using Substitution
Factoring Out the Greatest Common Factor
Families of Functions
Monomial Factors
Multiplying and Dividing Complex Numbers
Properties of Exponents
Multiplying Square Roots
Adding or Subtracting Rational Expressions with Different Denominators
Expressions with Variables as Exponents
The Quadratic Formula
Writing a Quadratic with Given Solutions
Simplifying Square Roots
Adding and Subtracting Square Roots
Adding and Subtracting Rational Expressions
Combining Like Radical Terms
Solving Systems of Equations Using Substitution
Dividing Polynomials
Graphing Functions
Product of a Sum and a Difference
Solving First Degree Inequalities
Solving Equations with Radicals and Exponents
Roots and Powers
Multiplying Numbers
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Use of Parentheses or Brackets (The Distributive Law)

Brackets as “Packages” for Negative Numbers

Sometimes brackets are used to avoid awkward-looking expressions involving negative numbers. For instance, to indicate the product of 5 and –3, we could write

(5) × (-3) or (5)(-3)

instead of

5 × –3

The brackets around the 5 in (5)(-3) don’t really mean anything, and could be dropped. By using brackets in this way, we can also drop the ‘×’ multiplication sign and just write (5)(-3) or 5(-3). Dropping the ‘×’ could not be done without the use of brackets, because then the already rather awkward expression ‘5 × –3’ would become ‘5 –3,’ which most people would interpret to mean “5 subtract 3” rather than “5 times –3”.


Distributing the Minus Sign

One of the trickier situations is when a pair of brackets is preceded by a minus sign. In such instances, regard the minus sign as meaning “multiply by –1.” Thus, for example,

-(5 – 3 + 6) = (-1) × (5 – 3 + 6)

= (-1) × [5 + (-3) + 6]

= (-1) × (5) + (-1) × (-3) + (-1) × (6)

= -5 + 3 – 6 = -8

In effect, when we remove brackets which are preceded by a minus sign, we need to reverse the signs of every term that was inside the brackets.

Notice how we used two different styles of brackets in one of the forms above so that matching pairs were easy to identify.


Nested Brackets

When the expression within a pair of brackets itself contains brackets, we say that the brackets are nested. In such situations, the removal of brackets must progress from the innermost pair of brackets to the outermost pair.

For example,

5 + 3[ 4 + 6(7 – 2) ]

= 5 + 3[ 4 + 6 × 5 ]

= 5 + 3[ 4 + 30 ]

= 5 + 3 × 34

= 5 + 102

= 107.

In the first step, we did the (7 – 2) = 5, since this pair of brackets was the innermost. This left just the square brackets as the only remaining pair of brackets (hence the innermost brackets), which we removed last.


A Caution

A little later in these notes, we will describe the general rule for deciding which operations to do in which order when faced with a more complication arithmetic or mathematical expression such as in the example above. Here we just mention a particular caution with respect to inserting brackets into an expression.

Sometimes brackets may be inserted into an expression for convenience:

5(6 – 2) = (5) (6) – (5) (2) = 5 × 6 – 5 × 2 = 30 – 10 = 20

The brackets in the first step have no effect except to act as visual separators for the numbers.

However, inserting brackets in an invalid manner causes an error – especially when addition or subtraction is mixed with multiplication. For example, it is a serious error to replace

4 + 5(6 – 2) with (4 + 5)(6 – 2).

We know that

4 + 5(6 – 2) = 4 + (5) (4) = 4 + 20 = 24.


(4 + 5) (6 – 2) = (9) (4) = 36,

a quite different number. The general rule for order of operations will state that multiplications must always be done before additions or subtractions. When we start with

4 + 5(6 – 2)

and insert brackets to get

(4 + 5)(6 – 2),

we are putting in brackets which force the leftmost addition to be done before the multiplication by 5. Thus, inserting the brackets to get this last expression violates the general rule we described that multiplications must be done before additions.

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