To compare numbers, we use the following symbols:

  • Greater than: “>”.
  • Less than: “<”.

For example: since 6 is greater than 2, we write 6>2. Similarly, since 2 is less than 6, we write 2<6.

We can use these symbols to describe characteristics of variables. For example, if we want to refer to numbers that are greater than 3, we write x>3. We call these expressions inequations.

Sometimes these types of expressions provide information indirectly. For example, '2x+1>0' indicates that twice x is greater than 1, but it is difficult to understand at first sight which are the numbers that fulfill this condition. To help us, we can solve the inequality (i.e. isolate the x) by doing some manipulations:

  • The expression '2x+1>0' indicates that the value of x meets the following: if we multiply it by 2 and add 1, the result has to be greater than 0. This only happens if 2x is greater than -1. Therefore, we can say that '2x>-1'.
  • So that twice x (2x) is greater than -1, it must happen that x is greater than -1/2. Therefore, we obtain ‘x>-1/2’.

The manipulations we have done are known as the “balance rule” and consist of doing the same operations on both sides of the sign (< or >):

  • In the first step, ‘2x+1>0; '2x>-1', we subtract 1 from both sides of the sign.
  • In the second step, ‘2x>-1; x>-1/2', we divide both by 2 sides of the sign.

Be careful! It is important to keep in mind that, if we multiply or divide both sides for a negative number, we need to change the ‘<’ sign to ‘>’ and reverse.

Other way to interpret the steps necessary to solve inequalities is through the “rules of the pass”:

  • What is adding happens by subtracting, and at reverse. In the first step, ‘2x+1>0; '2x>-1', the 1 passes by subtracting the other side of the sign.
  • What is multiplying happens dividing, and vice versa. In the second step, ‘2x>-1; x>-1/2', the 2 passes dividing the other side of the sign.

Be careful! Just like before, when a negative number is multiplying or dividing, It is necessary to change the sign when it passes to the other side.


Last modified: Friday, 19 July 2024, 2:56 PM