How Come When You Divide By A Negative Number The Inequality Sign Changes

Question 203735: Why does the inequality sign change when both sides are multiplied or divided by a negative number? Does this happen with equations? Why or why not? Write an inequality for your. In your inequality, use both the multiplication and add-on properties of inequalities.

Answer by jsmallt9(3758)

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Y'all can put this solution on YOUR website!
An inequality states that one number is less than (or greater than) another. I think the best style to understand what happens when you lot multiply or divide an inequality past a negative number is to flick the situation, earlier and later, on the number line

Motion picture any two numbers on the number line. The 1 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 aught and any number.

When you multiply (or separate) 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 1 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 at present 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 correct (i.east. greater than) ends upward to the left of (or less than) the other number.) This is why we have to reverse ("flip") the inequality whenever we multiply or split up it by a negative number.

If you're still having trouble with this, endeavour the post-obit:

Option any two (unlike) numbers.
Write the two numbers on the aforementioned line with some infinite between them.
Choice any negative number
Multiply or dissever (your choice) both of the commencement 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 betwixt the 2 numbers and write it there. (if y'all have trouble with this, plot the numbers on a number line. The i on the left is less than and the i on the correct is greater than.)
Echo this until you are comfortable with the fact that no thing what two dissimilar numbers y'all start with, no affair what negative number you use and no matter whether yous multiply or divide, the inequality always reverses!

Hither'south an example:

              Starting numbers: -8 and -4    Negative number: -3    Operation: Multiply    Results after multiplication: 24 and 12    Inequalities:       -eight < -iv       24 > 12

Does this happen with equalities (equations)? Yes, and no. The numbers volition withal flip over to the other side of zero. But the numbers were equal. They occupied the same betoken on the number line. After the numbers flip they will occupy the same new bespeak on the other side of zero and, therefore, they will still be equal afterward.

"Write an inequality for your. In your inequality, utilize both the multiplication and addition properties of inequalities." ???

Source: https://www.algebra.com/algebra/homework/Inequalities/Inequalities.faq.question.203735.html

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