You can put this solution on YOUR website!
An inequality states that one number is less than (or greater than) another. I think the best way to understand what happens when you multiply or divide an inequality by a negative number is to picture the situation, before and after, on the number line
two numbers on the number line. The one on the left is less than the one on the right. (The one on the right is greater than the one on the left.) This is true whether the two numbers are both positive, both negative, a negative and a positive or zero and any number.
When you multiply (or divide) any non-zero number by a negative, its sign changes. If it was positive it becomes negative and if it was negative it becomes positive. On a number line, the number "flips" over to the other side of zero. When you multiply (or divide) both sides of an inequality this happens to both numbers, the one on the "less than" side and the one on the "greater than" side. Both sides "flip". And when this happens the number that was on the left of the other before is now to the right of the other. In other words, what was less than before is now greater than. (And the number that was to the right (i.e. greater than) ends up to the left of (or less than) the other number.) This is why we have to reverse ("flip") the inequality whenever we multiply or divide it by a negative number.
If you're still having trouble with this, try the following:
- Pick any two (different) numbers.
- Write the two numbers on the same line with some space between them.
- Pick any negative number
- Multiply or divide (your choice) both of the first two numbers by this negative number and write the answers on another line with some space between them.
- Now, on each line, decide which inequality symbol belongs between the two numbers and write it there. (if you have trouble with this, plot the numbers on a number line. The one on the left is less than and the one on the right is greater than.)
- Repeat this until you are comfortable with the fact that no matter what two different numbers you start with, no matter what negative number you use and no matter whether you multiply or divide, the inequality always reverses!
Here's an example:
Starting numbers: -8 and -4
Negative number: -3
Results after multiplication: 24 and 12
-8 < -4
24 > 12
Does this happen with equalities (equations)? Yes, and no. The numbers will still flip over to the other side of zero. But the numbers were equal. They occupied the same point on the number line. After the numbers flip they will occupy the same new point on the other side of zero and, therefore, they will still be equal afterwards.
"Write an inequality for your. In your inequality, use both the multiplication and addition properties of inequalities." ???