Solving inequalities with "x" in the denominator has always been a stumbling block for me. Other than understanding how a particular expression, such as 1/x, works (in this case, x cannot be zero), how might I go about solving inequalities having such expressions mathematically.
1/x < 0 x(1/x) < x(0) 1 < 0 // This is where things seem to break down. // Probably because of the vertical asymptote.
As a slightly longer example:
1/x < 4 1/x - 4 < 0 x(1/x - 4) < x(0) 1 - 4x < 0 -4x < -1 x > 1/4 // Makes sense up to this point (Only covers x > 0) // How might I now solve for a negative x value?
I attempted changing the LessThan operator to a GreaterThan operator as per the rule:
if a < b and c < 0, then ac > bc
yet this resulted in an incorrect answer.
I already know that the answer to the inequality is (-infinity, 0) union (1/4, infinity). I would just like to know how to solve this algebraically.