Sure! Let's work through this step by step.
### Part (i): Verifying [tex]\( m - n \neq n - m \)[/tex]
Given values:
[tex]\[ m = -\frac{7}{9} \][/tex]
[tex]\[ n = \frac{5}{6} \][/tex]
First, let's calculate [tex]\( m - n \)[/tex]:
[tex]\[ m - n = -\frac{7}{9} - \frac{5}{6} \][/tex]
To subtract these fractions, we need a common denominator. The least common multiple (LCM) of 9 and 6 is 18.
Convert each fraction to have the denominator 18:
[tex]\[ -\frac{7}{9} = -\frac{7 \times 2}{9 \times 2} = -\frac{14}{18} \][/tex]
[tex]\[ \frac{5}{6} = \frac{5 \times 3}{6 \times 3} = \frac{15}{18} \][/tex]
Now subtract these fractions:
[tex]\[ m - n = -\frac{14}{18} - \frac{15}{18} = -\frac{14 + 15}{18} = -\frac{29}{18} \][/tex]
Next, let's calculate [tex]\( n - m \)[/tex]:
[tex]\[ n - m = \frac{5}{6} - (-\frac{7}{9}) \][/tex]
Rewriting [tex]\(-(-\frac{7}{9})\)[/tex]:
[tex]\[ n - m = \frac{5}{6} + \frac{7}{9} = \frac{5 \times 3}{6 \times 3} + \frac{7 \times 2}{9 \times 2} = \frac{15}{18} + \frac{14}{18} = \frac{29}{18} \][/tex]
We obtained:
[tex]\[ m - n = -\frac{29}{18} \][/tex]
[tex]\[ n - m = \frac{29}{18} \][/tex]
Clearly, [tex]\( m - n \)[/tex] and [tex]\( n - m \)[/tex] are not equal, thus:
[tex]\[ m - n \neq n - m \][/tex]
Verification (i) is true.
### Part (ii): Verifying [tex]\( -(m + n) = (-m) + (-n) \)[/tex]
First, compute [tex]\( m + n \)[/tex]:
[tex]\[ m + n = -\frac{7}{9} + \frac{5}{6} \][/tex]
Using the common denominator 18:
[tex]\[ -\frac{7}{9} = -\frac{14}{18} \][/tex]
[tex]\[ \frac{5}{6} = \frac{15}{18} \][/tex]
Add these fractions:
[tex]\[ m + n = -\frac{14}{18} + \frac{15}{18} = \frac{-14 + 15}{18} = \frac{1}{18} \][/tex]
Now, compute [tex]\( -(m + n) \)[/tex]:
[tex]\[ -(m + n) = -\left( \frac{1}{18} \right) = -\frac{1}{18} \][/tex]
Next, compute [tex]\( (-m) + (-n) \)[/tex]:
[tex]\[ -m = -\left( -\frac{7}{9} \right) = \frac{7}{9} \][/tex]
[tex]\[ -n = -\left( \frac{5}{6} \right) = -\frac{5}{6} \][/tex]
Using the common denominator 18 again:
[tex]\[ \frac{7}{9} = \frac{14}{18} \][/tex]
[tex]\[ -\frac{5}{6} = -\frac{15}{18} \][/tex]
[tex]\[ (-m) + (-n) = \frac{14}{18} - \frac{15}{18} = \frac{14 - 15}{18} = -\frac{1}{18} \][/tex]
Therefore:
[tex]\[ -(m + n) = -\frac{1}{18} \][/tex]
[tex]\[ (-m) + (-n) = -\frac{1}{18} \][/tex]
Since both are equal:
[tex]\[ -(m + n) = (-m) + (-n) \][/tex]
Verification (ii) is true.
In summary:
(i) [tex]\( m - n \neq n - m \)[/tex] is verified as true.
(ii) [tex]\( -(m + n) = (-m) + (-n) \)[/tex] is verified as true.
