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Saturday 13th of July
Calculations with Negative Numbers
Solving Linear Equations
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Algebra Expressions
Evaluating Expressions and Solving Equations
Fraction rules
Factoring Quadratic Trinomials
Multiplying and Dividing Fractions
Dividing Decimals by Whole Numbers
Adding and Subtracting Radicals
Subtracting Fractions
Factoring Polynomials by Grouping
Slopes of Perpendicular Lines
Linear Equations
Roots - Radicals 1
Graph of a Line
Sum of the Roots of a Quadratic
Writing Linear Equations Using Slope and Point
Factoring Trinomials with Leading Coefficient 1
Writing Linear Equations Using Slope and Point
Simplifying Expressions with Negative Exponents
Solving Equations 3
Solving Quadratic Equations
Parent and Family Graphs
Collecting Like Terms
nth Roots
Power of a Quotient Property of Exponents
Adding and Subtracting Fractions
Solving Linear Systems of Equations by Elimination
The Quadratic Formula
Fractions and Mixed Numbers
Solving Rational Equations
Multiplying Special Binomials
Rounding Numbers
Factoring by Grouping
Polar Form of a Complex Number
Solving Quadratic Equations
Simplifying Complex Fractions
Common Logs
Operations on Signed Numbers
Multiplying Fractions in General
Dividing Polynomials
Higher Degrees and Variable Exponents
Solving Quadratic Inequalities with a Sign Graph
Writing a Rational Expression in Lowest Terms
Solving Quadratic Inequalities with a Sign Graph
Solving Linear Equations
The Square of a Binomial
Properties of Negative Exponents
Inverse Functions
Rotating an Ellipse
Multiplying Numbers
Linear Equations
Solving Equations with One Log Term
Combining Operations
The Ellipse
Straight Lines
Graphing Inequalities in Two Variables
Solving Trigonometric Equations
Adding and Subtracting Fractions
Simple Trinomials as Products of Binomials
Ratios and Proportions
Solving Equations
Multiplying and Dividing Fractions 2
Rational Numbers
Difference of Two Squares
Factoring Polynomials by Grouping
Solving Equations That Contain Rational Expressions
Solving Quadratic Equations
Dividing and Subtracting Rational Expressions
Square Roots and Real Numbers
Order of Operations
Solving Nonlinear Equations by Substitution
The Distance and Midpoint Formulas
Linear Equations
Graphing Using x- and y- Intercepts
Properties of Exponents
Solving Quadratic Equations
Solving One-Step Equations Using Algebra
Relatively Prime Numbers
Solving a Quadratic Inequality with Two Solutions
Operations on Radicals
Factoring a Difference of Two Squares
Straight Lines
Solving Quadratic Equations by Factoring
Graphing Logarithmic Functions
Simplifying Expressions Involving Variables
Adding Integers
Factoring Completely General Quadratic Trinomials
Using Patterns to Multiply Two Binomials
Adding and Subtracting Rational Expressions With Unlike Denominators
Rational Exponents
Horizontal and Vertical Lines
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Adding Integers

Objective Learn how to add positive and negative integers.

You will learn how to add, subtract, multiply, and divide integers.

Rules for addition of negative numbers

  • Adding a negative number to a positive number gives the same result as subtracting the corresponding positive number.
9 + (-4) = 9 - 4
  = 5
  • When we subtract a larger number from a smaller one, the result will be a negative number.

5 - 7 = -2

  • Therefore, when we have an addition problem involving one positive number and one negative number, we may end up with a negative answer.
5 + (-7) = 5 - 7   14 + (-21) = 14 - 21
  = -2     = -7
  • If we add two negative numbers, then the result is the opposite of what we would get if we added the corresponding positive numbers:
(-2) + (-3) = -2 -3
  = -5

You should try to figure out why these rules are true. The following may help you:


Why are these rules for addition true?

You can represent positive numbers by collections of + 1 blocks and negative numbers by collections of - 1 blocks.

These collections of blocks represent - 6 and 8, respectively. We can also combine collections of positive (+1) and negative (-1) blocks. Think about this as adding a positive and a negative number.

This collection represents 8 + ( - 6).

Remember that we are thinking of negative numbers as a deficit, so the collections we just drew should be thought of as a collection of 8 (dollars, say) together with a deficit of 6 dollars. Of course, the 8 dollars could be used to pay off the deficit of 6 dollars, and there would be 2 dollars left over. This can be pictured using blocks if we allow ourselves to cancel a negative (-1) block and a positive (+1) block.

In the picture, we cancelled each negative (-1) block with a positive (+1) block and removed both blocks from the picture. Whenever we have both negative (-1) and positive (+1) blocks, we should cancel unlike blocks in pairs until we have a collection that consists only of blocks of a single color (kind).

Now consider 6 + ( - 8).

When we cancel pairs of positive (+1) and negative (-1) blocks, we are left with two negative (-1) blocks.

This means that we have a deficit of 2, so 6 + ( -8) = -2.


Example 4

Compute ( - 4) + ( - 6).


Draw -4 as a collection of four negative (-1) blocks and -6 as a collection of 6 negative (-1) blocks. When we merge them to add, we will get 10 negative (-1) blocks.

Therefore, ( - 4) + ( - 6) = -10.

Here is a very important special case of these rules of addition.

Adding Opposites When we add a positive number and its opposite negative number, we always get zero.


Example 5

Demonstrate that 5 + ( - 5) = 0.


Represent 5 + ( - 5) with 5 positive (+1) blocks and 5 negative (-1) blocks. The opposite blocks (-1 and +1) cancel in pairs. We are left with no blocks at all, so the result is 0.



Negative Numbers in Algebra

When working with variables, one can use the Distributive Property together with the rules about adding and subtracting integers to simplify expressions and equations.

Example 6

Simplify the expression 4x + ( - 3)x.


Use the distributive property.

4x + ( - 3)x = [4 + ( -3)] x
  = [1] x
  = x

As with actual numbers, one can think of each x and - x as units that cancel out, so -3x cancels out 3x .

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