Solve the equation 0.98x + 22.54 = 0.99x.

Step-by-step solution

1. 1

   Gather the x terms. Subtract $0.98x$ from both sides so the remaining coefficient stays positive:

   $22.54 = 0.99x - 0.98x = 0.01x$

   The whole equation has collapsed to a single term — the two near-equal percentages left only their $1\%$ difference behind.

2. 2

   Divide by the small coefficient.

   $x = \frac{22.54}{0.01} = 2254$

   Dividing by $0.01$ is the same as multiplying by $100$, which is why a modest constant produces a root in the thousands.

3. 3

   Work it as an exact fraction to be safe with decimals. $\tfrac{22.54}{0.01} = \tfrac{2254}{1}$ once both numerator and denominator are multiplied by $100$, so the answer is exact, not a rounded decimal.

4. 4

   Check by substituting back. Left: $0.98 \times 2254 + 22.54 = 2208.92 + 22.54 = 2231.46$. Right: $0.99 \times 2254 = 2231.46$. Identical.

5. 5

   Note the sensitivity. If the coefficients had been $0.98$ and $0.999$, the divisor would be $0.019$ and the root would fall to about $1186$. Equations where two coefficients nearly cancel are extremely sensitive to those coefficients, so they should be kept exact rather than rounded.