Hints for solving the problems - Apply 1.3

Problem No. 2  –  Find:    -22 + 10

Remember when adding signed numbers:

Like signs add the absolute values and keep the same sign.

Unlike signs subtract the absolute values and keep the sign of the largest

absolute value.

     -22 + 10 = 22 minus 10 = 12, keep the negative sign.   The answer is -12.

Also when subtracting signed numbers change the sign of the subtrahend and use the rules of addition. The subtrahend is the one on the right or the one below.

Problem No. 6  –  Find:    -9 – (-36)

This is a subtraction problem, because there is a negative sign in front of the parentheses. Problem No. 2 was an addition problem. The two signs in front of the numbers are signs of opposition (positive and negative). Problem No. 6 the signs in front of the 9 and 36 are signs of opposition, but the sign in front of the parentheses is called a sign of operations. This operation is shown as (-) which is subtraction. If there was a plus (+) sign in front of the parentheses, the problem would be an addition problem.

Problem such as: 6 – 8 + 5 – 4 + 2 – 1 show the operations of addition. Yes you will subtract some of the absolute values, but the original problem doesn’t show any subtraction symbols.

In problem No. 6 change the sign of the 36 and use the rules of addition.

        -9 + 36 = 27   The answer is 27.

Problem No. 8  –  Find:     -6 · 24

In multiplication problems, multiply the absolute values and:

If the signs are the same, the answer will be positive.

If the signs are different, the answer will be negative.

                   -6 · 24 = - 144   The answer is -144.

Problem No. 12  –  Find:     -36 · (-18)

Multiply the absolute values and the answer will be positive, because the original numbers had the same sign. (Both negative)

         -36 · (-18) = 648     The answer is 648.

Problem No. 14  –  Find:     -136 ÷ 8

In division problems, divide the absolute values and:

If the signs are the same, the answer will be positive.

If the signs are different, the answer will be negative.

                -136 ÷ 8 = -17    The answer is -17.

Problem No. 22  –  Find:    54 ÷ [3 – (2 · 3)]

To solve this problem, you need to know the order of operations.

Order of operations

        3. Multiply or divide, working from left to right.

        4. Add or subtract, working from left to right.

First step in this problem, multiply the 2 times the 3.

             54 ÷ [3 – (6)]

Next, subtract the 3 and the 6.

              54 ÷ [-3]

Next, divide the 54 and the 3.

              54 ÷ [-3] = -18     The answer is -18.

Problem No. 27 – Find:      (-20 + 4) ÷ 2³ - 3 · 4²

First step, add the -20 and 4

               -16 ÷ 2³ - 3 · 4²

Next, you may raise the 2 and 4 to their powers.

                -16 ÷ 8 – 3 · 16

Now remember multiplication and division is done in order from left to right, so you divide the -16 by 8.

                  -2 - 3 · 16

Next, multiply the -3 times 16.

                  -2 - 48 = -50    The answer is -50.

Commutative Property of Addition and Multiplication

Associative Property of Addition and Multiplication

Distributive Property

Additive and Multiplicative Identity

Additive and Multiplicative Inverse

Commutative Property of Addition and Multiplication

             4 + 5 = 5 + 4 4(5) = 5(4)

It makes no difference in the order when you add or multiply two numbers.

Remember these operations are binary, as you add or multiply only two numbers at the same time and then to that answer you add or multiply the next number.

Subtraction and Division are not Commutative.

        Does 4 - 5 = 5 - 4      no

        Does 4 ÷ 2 = 2 ÷ 4    no

Associative Property of Addition and Multiplication

          4 + (5 + 6) = (4 + 5) + 6 4(5 · 6) = (4 · 5)6

When adding or multiplying numbers, it doesn’t make any difference which two are added or multiplied first. Then that answer is added or multiplied to the other number.

Subtraction and Division not Associative.

Distributive Property

             4(x + 3) = 4x + 12 3(x - 6) = 3x - 18

The number or variable is multiplied by both terms in the parentheses. If the problem has a plus sign it is multiplication over addition. If the problem has a negative sign it is multiplication over subtraction.

This property will be used the most in Algebra.

Additive and Multiplicative Identity

           5 + 0 = 5 5(1) = 5

This property shows that when you add zero to any number you get the same number that you started with.

It also shows what when you multiply any number by one you get the same number that you started with.

      0 is the additive identity element.

      1 is the multiplicative identity element.

Additive and Multiplicative Inverse

        5 + (-5) = 0 5(1/5) = 1

This property shows that when you add any number to its opposite you get 0.

Also when you multiply any number by it’s inverse (reciprocal) you get 1.

Given this problem, determine the property that justifies it.

         5(6 – 8) + 2 = 5 · 6 + 5 · (-8) + 2

What changes from the left side of the equation to the right side of the equation? The 5 is multiplied by the 6 and the -8.

So this is the distributive property.