Which mathematical operation is restricted in the context of working with inequalities?

In the context of working with inequalities, multiplying both sides of an inequality by a negative number is the operation that is restricted. When you multiply or divide both sides of an inequality by a negative number, the direction of the inequality sign must be reversed. This is crucial because it alters the meaning of the inequality.

For example, if you have the inequality ( x < 3 ) and you multiply both sides by -1, it transforms into ( -x > -3 ). This reversal can lead to confusion and incorrect assumptions if one does not remember to switch the direction of the inequality.

Adding or subtracting a constant from both sides of an inequality does not change its direction and is therefore always permissible. Similarly, when multiplying or dividing by a positive number, the inequality remains the same. Thus, it is specifically the case of multiplying by a negative number that creates the restriction on operations with inequalities.