A-Level MathsInequalities

Things you need to know about inequalities:

  • If the question has an inequality such as \leq,\hspace{2pt}\geq, \hspace{2pt}\textless, \hspace{2pt}\textgreater, then the solution must include inequalities and not equals signs. This is a common error that students make.
  • Linear inequalities can be solved in very much the same way as linear equations. See Example 1.
  • Quadratic inequalities can be solved by finding the roots of the quadratic, sketching the graph having knowledge of the roots and identifying where the graph is positive or where it is negative. If the graph has two separate regions then the answer must have two separate regions. See Example 2.
  • You can plot regions on the number line and so if you want to see where two inequalities hold (simultaneous inequalities) then you can see where the two solution regions overlap on the number line. See Example 3.

NOTE the following very important facts about inequalities in addition to the above:

  1. Multiplying or dividing both sides of an inequality by a negative number reverses the sign of the inequality.
  2. Taking the reciprocal of both sides of an inequality also reverses the sign of the inequality.

See Example 4.

Example 1

Solve the inequality 2x+3\hspace{2pt}\geq \hspace{2pt}5x-9.

Example 2

Solve the quadratic inequality x^2-x\hspace{2pt}\textgreater\hspace{2pt} 6.

Example 3

Find the region where both the inequalities 2x+3\hspace{2pt}\geq \hspace{2pt}5x-9 and x^2-x\hspace{2pt}\textgreater\hspace{2pt} 6 both hold.

Example 4

Solve the following inequalities:  

  1. -x\leq 4
  2. \frac{1}{x}\hspace{2pt}\textgreater \hspace{2pt}7