Answer:
Let us consider any two numbers 6 and 9 then we have the relation.
[tex] \gray{ \frak{9 > 6 \: \: \: or \: \: \: 6 < 9}}[/tex]
Rule 1.
Adding 2 on both sides then;
[tex] \frak{ \gray{9 + 2 > 6 + 2 \: \: \: or \: \: \: 6 + 2 < 9 + 2}}[/tex]
[tex]\frak{ \gray{ \therefore \: 11 > 8 \: \: \: or \: \: \: 8 < 11}}[/tex]
Both are true for addition.
Thus,when equal number is added to both sides of trichotomy symbol,the symbol remains the same.
Rule 2.
Subtracting 2 on the both sides then,
[tex]\frak{ \gray{9 - 2 > 6 - 2 \: \: \: or \: \: \: 6 - 2 < 9 - 2}}[/tex]
[tex]\frak{ \gray{\therefore \: 7 > 4 \: \: \: or \: \: \: 4 < 7}}[/tex]
Both are true for subtraction.
Thus,when equal number is subtracted to both sides of trichotomy symbol,the symbol remains the same.
Rule 3.
Multiplying both sides by 2 then,
[tex]\frak{ \gray{9 \times 2 > 6 \times 2 \: \: \: or \: \: \: 6 \times 2 < 9 \times 2}}[/tex]
[tex]\frak{ \gray{ \therefore \: 18 > 12 \: \: \: or \: \: \: 12 < 18}}[/tex]
Both are true for multiplication by positive numbers.
But if we multiplied by -2 then there will be,
[tex]\frak{ \gray{ - 18 > - 12 \: \: \: or \: \: \: - 12 < - 18}}[/tex]
These are not true.
To be true,there must be:
[tex]\frak{ \gray{ - 18 < - 12 \: \: \: and \: \: \: - 12 > - 18}}[/tex]
Thus,when both sides of trichotomy symbol are multiplied by equal positive number then the symbol remains the same.When both sides of trichotomy symbols are multiplied by equal negative numbers then the symbol changes as follows:
'<' Changes to'>' and '>' Changes to'<'.
Similarly check for division then get the same result as in multiplication.
Thus,when both sides of trichotomy symbol are divided by an equal positive number, the symbol remains the same.When both sides of trichotomy symbol are divided by an equal negative number,the symbol '<' is changed to'>' or '>' is changed to'<'.
[tex]\boxed{\frak{\gray{BrainlyFairy}}}[/tex]
