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

1 Answer
Aug 12, 2015

Answer:

#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#