Answered

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7. If [tex]m=-\frac{7}{9}[/tex] and [tex]n=\frac{5}{6}[/tex], verify that:

(i) [tex]m-n \neq n-m[/tex]

(ii) [tex]-(m+n)=(-m)+(-n)[/tex]

Sagot :

GinnyAnswer
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.
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