Solving equations and inequalities involving 1/x

Equations of the form 1/x = k

To solve 1/x = k (where k is not equal to 0), multiply both sides by x (where x is not equal to 0):

1 = kx -> x = 1/k

Example: 1/x = 3 gives x = 1/3.

Note: if k = 0, the equation 1/x = 0 has NO solution, as 1/x is never zero, regardless of the value of x.

Equations of the form 1/x = 1/a

If 1/x = 1/a (where a is non-zero), then x = a is the only solution, provided that x remains non-zero.

Inequalities of the form 1/x > 0

The sign of 1/x is the same as that of x. Therefore:

• 1/x > 0 is equivalent to x > 0
• 1/x < 0 is equivalent to x < 0

Inequalities of the form 1/x > k (trap!)

To solve 1/x > 2, for example, you CANNOT multiply by x directly without knowing its sign: the direction of the inequality depends on the sign of x. The safe method is to reduce it to a single fraction compared with 0:

1/x - 2 > 0 -> (1 - 2x)/x > 0

We then examine the signs of the numerator and denominator separately in a sign table, and conclude the sign of the quotient.

Common pitfall

Never multiply an inequality by x without knowing whether x is positive or negative: the direction of the inequality changes depending on the sign. You must always reduce the inequality to a single quotient compared with 0 before drawing a conclusion.