Evaluate \(1 - \left(\frac{1}{5} + 1\frac{2}{3}\right) + \left(5 + 1\frac{2}{3}\right)\)

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$1?(\frac{1}{5}\times \frac{2}{3})+(5+\frac{2}{3})1\; -\; \backslash left(\; \backslash frac\{1\}\{5\}\; \backslash times\; \backslash frac\{2\}\{3\}\; \backslash right)\; +\; \backslash left(\; 5\; +\; \backslash frac\{2\}\{3\}\; \backslash right)$

Step 1: Evaluate $\frac{1}{5}\times \frac{2}{3}\backslash frac\{1\}\{5\}\; \backslash times\; \backslash frac\{2\}\{3\}$

$\frac{1}{5}\times \frac{2}{3}=\frac{1\times 2}{5\times 3}=\frac{2}{15}\backslash frac\{1\}\{5\}\; \backslash times\; \backslash frac\{2\}\{3\}\; =\; \backslash frac\{1\; \backslash times\; 2\}\{5\; \backslash times\; 3\}\; =\; \backslash frac\{2\}\{15\}$

51?×32?=5×31×2?=152?

Step 2: Subtract $\frac{2}{15}\backslash frac\{2\}\{15\}$ from 1

$1?\frac{2}{15}=\frac{15}{15}?\frac{2}{15}=\frac{13}{15}1\; -\; \backslash frac\{2\}\{15\}\; =\; \backslash frac\{15\}\{15\}\; -\; \backslash frac\{2\}\{15\}\; =\; \backslash frac\{13\}\{15\}$

1?152?=1515??152?=1513?

Step 3: Simplify $5+\frac{2}{3}5\; +\; \backslash frac\{2\}\{3\}$

First, convert $55$ to a fraction with a denominator of 3:

$5=\frac{15}{3}5\; =\; \backslash frac\{15\}\{3\}$

So,

$5+\frac{2}{3}=\frac{15}{3}+\frac{2}{3}=\frac{17}{3}5\; +\; \backslash frac\{2\}\{3\}\; =\; \backslash frac\{15\}\{3\}\; +\; \backslash frac\{2\}\{3\}\; =\; \backslash frac\{17\}\{3\}$

Step 4: Combine the results

$\frac{13}{15}+\frac{17}{3}\backslash frac\{13\}\{15\}\; +\; \backslash frac\{17\}\{3\}$

To combine these fractions, find a common denominator. The least common multiple of 15 and 3 is 15:

Convert $\frac{17}{3}\backslash frac\{17\}\{3\}$ to a fraction with a denominator of 15:

$\frac{17}{3}=\frac{17\times 5}{3\times 5}=\frac{85}{15}\backslash frac\{17\}\{3\}\; =\; \backslash frac\{17\; \backslash times\; 5\}\{3\; \backslash times\; 5\}\; =\; \backslash frac\{85\}\{15\}$

Now add the fractions:

$\frac{13}{15}+\frac{85}{15}=\frac{13+85}{15}=\frac{98}{15}\backslash frac\{13\}\{15\}\; +\; \backslash frac\{85\}\{15\}\; =\; \backslash frac\{13\; +\; 85\}\{15\}\; =\; \backslash frac\{98\}\{15\}$

Simplify the fraction if possible:

$\frac{98}{15}=6\frac{8}{15}\backslash frac\{98\}\{15\}\; =\; 6\; \backslash frac\{8\}\{15\}$

Thus, the evaluated expression is:

$\boxed{{\displaystyle 6\frac{8}{15}}}\backslash boxed\{6\; \backslash frac\{8\}\{15\}\}$