What If I Solve It, But "x" Is On The Right?

No matter, just swap sides, but reverse the sign so it still "points at" the correct value!


Note: "x" can be on the right, but people usually like to see it on the left hand side.

You are watching: If you multiply an inequality by a negative number

Multiplying or Dividing by a Value

Another thing we do is multiply or divide both sides by a value (just as in Algebra - Multiplying).

But we need to be a bit more careful (as you will see).

Positive Values

Everything is fine if we want to multiply or divide by a positive number:


Example: 3y
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When we multiply or divide by a negative number we must reverse the inequality.

Why?

Well, just look at the number line!

For example, from 3 to 7 is an increase, but from −3 to −7 is a decrease.

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See how the inequality sign reverses (from ) ?


Multiplying or Dividing by Variables

Here is another (tricky!) example:


Example: bx 3

But we don"t know if b is positive or negative, so we can"t answer this one!


To help you understand, imagine replacing b with 1 or −1 in the example of bx if b is 1, then the answer is x but if b is −1, then we are solving −x 3

The answer could be x 3 and we can"t choose because we don"t know b.

See more: How To Circumscribe A Circle Around A Triangle S &Mdash; Krista King Math

So:

Do not try dividing by a variable to solve an inequality (unless you know the variable is always positive, or always negative).


A Bigger Example


Example: x−32 x−32 ×2 6−2x3 > x > −3

And that is the solution!

But to be neat it is better to have the smaller number on the left, larger on the right. So let us swap them over (and make sure the inequalities point correctly):

−3 x 6


Summary

Many simple inequalities can be solved by adding, subtracting, multiplying or dividing both sides until you are left with the variable on its own. But these things will change direction of the inequality: Multiplying or dividing both sides by a negative number Swapping left and right hand political parties Don"t multiply or divide by a variable (unless you know it is always positive or always negative)