# How do you solve 90< -6w?

Aug 12, 2015

$w < - 15$

#### 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:
$2 < 4$ $\implies$ $2 \cdot 10 < 4 \cdot 10$
$2 < 4$ $\implies$ $2 \cdot \frac{1}{4} < 4 \cdot \frac{1}{4}$
$- 8 < 4$ $\implies$ $- 8 \cdot 10 < 4 \cdot 10$
$- 8 < 4$ $\implies$ $- 8 \cdot \frac{1}{4} < 4 \cdot \frac{1}{4}$
$- 8 < - 4$ $\implies$ $- 8 \cdot 10 < - 4 \cdot 10$
$- 8 < - 4$ $\implies$ $- 8 \cdot \frac{1}{4} < - 4 \cdot \frac{1}{4}$

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:
$2 < 4$ $\implies$ $2 \cdot \left(- 10\right) > 4 \cdot \left(- 10\right)$
$2 < 4$ $\implies$ $2 \cdot \left(- \frac{1}{4}\right) > 4 \cdot \left(- \frac{1}{4}\right)$
$- 8 < 4$ $\implies$ $- 8 \cdot \left(- 10\right) > 4 \cdot \left(- 10\right)$
$- 8 < 4$ $\implies$ $- 8 \cdot \left(- \frac{1}{4}\right) > 4 \cdot \left(- \frac{1}{4}\right)$
$- 8 < - 4$ $\implies$ $- 8 \cdot \left(- 10\right) > - 4 \cdot \left(- 10\right)$
$- 8 < - 4$ $\implies$ $- 8 \cdot \left(- \frac{1}{4}\right) > - 4 \cdot \left(- \frac{1}{4}\right)$

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 $- \frac{1}{6}$. As a result, the right side will be equal to $- 6 w \cdot \left(- \frac{1}{6}\right) = w$, while the left side will be equal to $90 \cdot \left(- \frac{1}{6}\right) = - 15$. Since we multiplied by a negative number $- \frac{1}{6}$, 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
$w < - 15$