Explain the difference between the multiplication property of equality and the multiplication property for inequalities. Give examples to support your response

Answer 1

Multiplication Property of EQUALITY applies to EQUATIONS,

and states that both sides of the equation can be multiplied

by the same non-zero number.

Multiplication Property of INEQUALITY is exactly the same,

with the additional stipulation is that if the number multiplying

both sides is NEGATIVE, then the inequality must be reversed.

The reason for this is easy to see....

For example , starts with 1 > 0.

Upon multiplying by -1, the inequality becomes -1 > 0,

which is FALSE; hence the reversal of the inequality is required.

Proof: for real numbers x and y...

Case 1: x>y > 0. Then -x<-y<0

Case 2: x>0 > y. then -x < 0 < -y

Case 3: x<y<0. then x>y>0

a number line will help visualize this..

For example, in Case 1: let x=5 and y=3. then -5<-3<0.

Case 2: let x=4 and y=-2. Then -4 < 0 < 2

case 3: let x=-6 and y=-3. Then 6>3>0

Here are several examples:

3x = 9 --> multiplies both sides by 1/3: 3 on left side

is cancelled; right side x = 1/3*9 = 3

(1/2)z = 5 --> multiplies both sides by 2: the 1/2 on left side

is cancelled; right side z = 2*5=10

-2 t = 10 --> multiplies both sides by -1/2: -2 on the left side

is cancelled; right side is (-1/2)*10 = -5

(3/4) y = 24 --> multiplies both sides by 4/3: 3/4 on the left side

is cancelled; right side is (4/3)*24 = 32

(-2/7) x = 14 --> multiplies both sides by -7/2: -2/7 on the left side

is cancelled; right side is (-7/2)*14 = -49

Now, here are example of INEQUALITIES based on the previous examples

of equations:

3x > 9 ---> x> 3

(1/2)z <=5 ---> z<=10

-2 t >= 10 ---> t <=-5 note that both sides are multiplied by negative

number, so the inequality sign must be reversed

(3/4)y > 24 ---> y > 32

(-2/7) x < 14 ---> x > -49 sign must be reversed as both sides are

multiplied by negative