Why can't we do crossing over in inequalities?
If we have the inequality #12/(x-4)<=4# , why can't we multiply both sides by #x-4# ?
If we have the inequality
1 Answer
Because you might be mutliplying both sides by a negative quantity.
Explanation:
Given any inequality, some operations that you can perform and preserve the truth or falsity of the inequality are:
- Add or subtract the same expression on both sides.
- Multiply or divide both sides by a positive expression.
- Multiply or divide both sides by a negative expression and reverse the inequality (i.e.
#<# becomes#># ,#>=# becomes#<=# , etc.). - Apply the same strictly monotonically increasing function to both sides.
- Apply the same strictly monotonically decreasing function to both sides and reverse the inequality.
Given:
#12/(x-4) <= 4#
we would like to get
One option is to note that
We find:
#12(x-4) <= 4(x-4)^2 = 4(x^2-8x+16)#
Then we can divide both ends by
#3x-12 <= x^2-8x+16#
Then subtract
#0 <= x^2-11x+28 = (x-4)(x-7)#
Note this inequality would be true for
but the point
So the solution of the original inequality is:
#x in (-oo, 4) uu [7, oo)#