Solve (x − 1) × 0.6 + 3 = 51203.3 for x, giving an exact value.

Step-by-step solution

1. 1

   Undo the addition first. Working outward from the constant term:

   $(x-1)\cdot 0.6 = 51203.3 - 3 = 51200.3$

   The bracket is treated as one block — there is no need to expand $0.6x - 0.6$ at this stage.

2. 2

   Divide by 0.6 as a fraction, not a decimal. Since $0.6 = \tfrac{3}{5}$, dividing by it means multiplying by $\tfrac{5}{3}$:

   $x - 1 = \frac{51200.3}{0.6} = \frac{512003}{10}\cdot\frac{5}{3} = \frac{512003}{6}$

   Going through fractions is what keeps the answer exact; $512003$ is not divisible by $3$ (its digits sum to $11$), so the fraction will not terminate as a decimal.

3. 3

   Add 1 to isolate x. Writing $1 = \tfrac66$:

   $x = \frac{512003}{6} + \frac{6}{6} = \frac{512009}{6}$

4. 4

   Convert to a decimal. Long division gives

   $x = \frac{512009}{6} = 85334.8\overline{3} \approx 85334.8333$

   The repeating $3$ is unavoidable, which is why the fractional form is the honest "exact answer".

5. 5

   Substitute back to verify exactly.

   $\left(\frac{512009}{6} - 1\right)\cdot\frac{3}{5} + 3 = \frac{512003}{6}\cdot\frac{3}{5} + 3 = \frac{512003}{10} + 3 = 51200.3 + 3 = 51203.3 \ \checkmark$

   The check closes with no rounding at all, which a decimal-only route could not demonstrate.