Solve for x: 8.8 - 10000x = 1010000x + 6000.

Step-by-step solution

1. 1

   Predict the answer's size before computing. The $x$ coefficients are huge ($10^{4}$ and $10^{6}$) while the constants are modest, so $x$ must be tiny. And since the left side has $6000$ to make up against only $8.8$, the $x$ term has to be negative — so expect a small negative number.

2. 2

   Collect the $x$ terms. Adding $10000x$ to both sides moves everything onto the right, keeping the coefficient positive:

   $8.8=1010000x+10000x+6000=1020000x+6000.$

   Adding rather than subtracting avoids a negative leading coefficient and the sign flip that goes with it.

3. 3

   Collect the constants. Subtracting $6000$:

   $8.8-6000=1020000x\;\Longrightarrow\;-5991.2=1020000x.$

   The left side is negative, matching the prediction in step 1.

4. 4

   Divide to isolate $x$. Dividing both sides by $1020000$:

   $x=\frac{-5991.2}{1020000}=-0.00587372549\ldots\approx-0.005874.$

   As an exact fraction, multiplying top and bottom by $10$ gives $x=-\tfrac{59912}{10200000}=-\tfrac{7489}{1275000}$.

5. 5

   Check by evaluating both sides. With $x=-0.00587372549$:

   $8.8-10000x=8.8+58.7372549=67.5372549,$

   $1010000x+6000=-5932.4627451+6000=67.5372549 ✓.$

   The two sides agree to eight decimal places, and the magnitude is exactly the "tiny and negative" that step 1 predicted.