Algebra Tutorials!

 Tuesday 30th of May

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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).

Solution

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.

Solution

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.

Solution

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 .