3. Solving Non-linear Inequalities

Another method of solving inequalities is to express the given inequality with zero on the right side and then determine the sign of the resulting function from either side of the root of the function.

The steps are as follows:

  1. Rewrite the inequality so that there is a zero on the right side.
  2. Find all linear factors of the function.
  3. To find the critical values, set each linear function to zero and solve for x.
  4. Determine the sign of the function in the intervals formed by the critical values.
  5. The solution will be those intervals in which the function has the correct signs satisfying the inequality.

Example 1

Solve the inequality x2 − 3 > 2x

Example 2

Solve the inequality x3 − 4x2 + x + 6 < 0

Example 3

Solve the inequality `((x-2)^2(x+3))/(4-x) < 0`


1. Solve `x^3-2x^2+x >= 0`

2. The mass m, in Mg, of fuel in a rocket after launch is

m = 2000 − t2 − 140t

where t is measured in minutes. During what time is the mass of fuel greater than `500\ "Mg"`?

3. The object distance p, and the image distance q, for a camera of focal length 3 cm is given by:


For what values of q is p > 12 cm?