Question

How do you solve `90< -6w`?

Answer 1

`w < -15`

Explanation:

Case 1
If you take two unequal real numbers and multiply both of them by the same positive real number, the relationship between them will be the same - the one that was greater before the multiplication will be greater after.

Here are a few examples:
`2 < 4` `=>` `2*10 < 4*10`
`2 < 4` `=>` `2*1/4 < 4*1/4`
`-8 < 4` `=>` `-8*10 < 4*10`
`-8 < 4` `=>` `-8*1/4 < 4*1/4`
`-8 < -4` `=>` `-8*10 < -4*10`
`-8 < -4` `=>` `-8*1/4 < -4*1/4`

We conclude with the following rule #1 about transformations of inequalities:
both sides of inequality can be multiplied by any positive real number without change in a sign of inequality.

Case 2
If you take two unequal real numbers and multiply both of them by the same negative real number, the relationship between them will change to opposite - the one that was greater before the multiplication will be less after.

Here are a few examples:
`2 < 4` `=>` `2*(-10) > 4*(-10)`
`2 < 4` `=>` `2*(-1/4) > 4*(-1/4)`
`-8 < 4` `=>` `-8*(-10) > 4*(-10)`
`-8 < 4` `=>` `-8*(-1/4) > 4*(-1/4)`
`-8 < -4` `=>` `-8*(-10) > -4*(-10)`
`-8 < -4` `=>` `-8*(-1/4) > -4*(-1/4)`

We conclude with the following rule #2 about transformations of inequalities:
both sides of inequality can be multiplied by any negative real number, but a sign of inequality should be changed to opposite.

In the problem at hand
`90 < −6w`
we can multiply both sides of the inequality by a negative number `-1/6`. As a result, the right side will be equal to `-6w*(-1/6)=w`, while the left side will be equal to `90*(-1/6)=-15`. Since we multiplied by a negative number `-1/6`, the sign of inequality should be changed to opposite, that is the resulting inequality will be
`-15 > w`.

Since it is customary to place the unknown variable (`w` in this case) on the left side of inequality, we can express the same relationship as
`w < -15`