Question

Tell if the following statement is true or false. In any case give a valid reason for saying so.
P: If x and y are integers such that \[x > y\] then \[ - x < - y\]
A. TRUE
B. FALSE

Answer 1

Hint: We use the concept of inequalities that when a negative sign is multiplied, it is multiplied to
both sides of the inequality and it changes the nature of the inequality.
* If we multiply -1 to an equation having inequality < or > then it changes to > or < respectively.
* The set of Integers consists of \[\left\{ {..... - 3, - 2, - 1,0,1,2,3....} \right\}\]

Complete step-by-step answer:
We are given two integers x and y
From the definition of integers we know they can be both negative, both positive or one negative and one positive.
We are given the inequality that holds true is \[x > y\] ………... (1)
We multiply the left hand side and right hand side of the inequality by -1
Then the sign of inequality changes from > to <
\[ \Rightarrow - 1 \times x < - 1 \times y\]
On multiplication of terms on both sides of the inequality we get
\[ \Rightarrow - x < - y\]
\[\therefore \]If we are given \[x > y\] then \[ - x < - y\]
\[\therefore \]Statement given in the question is TRUE.

So, option A is correct.

Note: Many students opt for proving the statement given by taking three separate cases each showing signs of integers and then multiplying -1 to each inequality in each case. This is a long process and it is not necessary as both numbers are given as integers, so any change in one integer will be easy to compare to another integer. Also, many students get confused why we multiply both sides of the equation by -1, they can think of the inequality as fraction \[\dfrac{x}{y} > 1\] and then if we multiply any value to numerator and same value to denominator then only the value of fraction remains the same otherwise it changes.