Evaluating the Repeating Decimal Expression

The problem asks us to evaluate the expression involving repeating decimals: $5.29\bar{8} - 6.29\bar{8} + 7.29\bar{8}$. To solve this, we will convert each repeating decimal into its equivalent fraction and then perform the subtraction and addition.

Converting Repeating Decimals to Fractions

A repeating decimal is a decimal number that has a digit or a sequence of digits that repeat infinitely. We can convert these into fractions using an algebraic method.

Converting $5.29\bar{8}$

1. Let $x = 5.29\bar{8}$.
2. Multiply by 100 to move the decimal point before the repeating part: $100x = 529.\bar{8}$.
3. Multiply by 1000 (100 times 10, since one digit repeats) to get the repeating part aligned: $1000x = 5298.\bar{8}$.
4. Subtract the equation from step 2 from the equation in step 3:

   $1000x - 100x = 5298.\bar{8} - 529.\bar{8}$

   $900x = 4769$

5. Solve for $x$:

   $x = \frac{4769}{900}$

Converting $6.29\bar{8}$

Using the same method:

1. Let $y = 6.29\bar{8}$.
2. $100y = 629.\bar{8}$
3. $1000y = 6298.\bar{8}$
4. Subtract:

   $1000y - 100y = 6298.\bar{8} - 629.\bar{8}$

   $900y = 5669$

5. Solve for $y$:

   $y = \frac{5669}{900}$

Converting $7.29\bar{8}$

Again, applying the same steps:

1. Let $z = 7.29\bar{8}$.
2. $100z = 729.\bar{8}$
3. $1000z = 7298.\bar{8}$
4. Subtract:

   $1000z - 100z = 7298.\bar{8} - 729.\bar{8}$

   $900z = 6569$

5. Solve for $z$:

   $z = \frac{6569}{900}$

Performing the Arithmetic Operation

Now substitute the fractional equivalents back into the original expression:

$5.29\bar{8} - 6.29\bar{8} + 7.29\bar{8} = x - y + z$

$= \frac{4769}{900} - \frac{5669}{900} + \frac{6569}{900}$

Calculation Steps

Since the denominators are the same, we can combine the numerators:

1. Combine the numerators:

   $ \frac{4769 - 5669 + 6569}{900} $

2. Perform the subtraction first: $4769 - 5669 = -900$.

   $ \frac{-900 + 6569}{900} $

3. Perform the addition: $-900 + 6569 = 5669$.

   $ \frac{5669}{900} $

Final Answer

The result of the evaluation $5.29\bar{8} - 6.29\bar{8} + 7.29\bar{8}$ is $\frac{5669}{900}$.