Does Multiplying or Dividing by a Negative Number Change the Inequality Symbol?

[xeon123]

When we multiply or divide by a negative number a inequality of the type ≤, the symbol will become ≥, or >?


-2x≥-4y, will become x ≤ 2y, or x < 2y?

 

[chiro]

When we multiply or divide by a negative number a inequality of the type ≤, the symbol will become ≥, or >?


-2x≥-4y, will become x ≤ 2y, or x < 2y?

Hey xeon123 and welcome to the forums.

In general under most normal transformations, if you have an equality, the equality after the transformation is maintained and this applies for your inequality example.

So basically its <= and not <.

Also for the same kind of example as above, strict inequality results in another strict inequality.

This isn't always the case, but if you are just doing standard arithmetic operations, then yeah a strict inequality remains an inequality and a non-strict inequality (that contains an equals) also will be a non-strict inequality after the operation.

 

[HallsofIvy]

Have you tried it with numbers? 2< 3, right? Now is -2< -3 or the other way around?

 

[chiro]

Have you tried it with numbers? 2< 3, right? Now is -2< -3 or the other way around?

That's not what he is asking: he is asking if a strict inequality goes to a strict inequality under an arithmetic operation. So basically multiply by negative makes >= to <= instead of >= to <.

 

[CellCoree]

When multiplying or dividing by a negative number, the direction of the inequality symbol will indeed change. In the given example, -2x≥-4y, when we divide both sides by -2, the inequality becomes x≤2y. This is because when we divide by a negative number, the direction of the inequality is reversed. So, -2x is larger than -4y, but when we divide by -2, x becomes smaller than 2y.

Similarly, if we have an inequality such as -3x>12, when we divide both sides by -3, the inequality becomes x<-4. This is because -3x is greater than 12, but when we divide by -3, x becomes smaller than -4.

In summary, when we multiply or divide by a negative number, the direction of the inequality symbol will change. This is important to keep in mind when solving inequalities and graphing them on a number line.