Solve 5.28 × (3.64x − 9.71) + (x + 8.12)/4.79 = −(9.46x − 2.53)/7.01, giving x correct to two decimal places.

Step-by-step solution

1. 1

   Expand the bracketed product. Multiply the decimals exactly, keeping full precision until the very end — rounding early is the main source of error in this kind of exercise:

   $5.28\times 3.64=19.2192,\qquad 5.28\times 9.71=51.2688$

   so the first term is $19.2192x-51.2688$.

2. 2

   Split each fraction into an x-part and a constant part. Dividing a sum term by term turns the fractions into ordinary decimal coefficients:

   $\frac{x+8.12}{4.79}=\frac{1}{4.79}x+\frac{8.12}{4.79}=0.208768x+1.695198$

   $-\frac{9.46x-2.53}{7.01}=-\frac{9.46}{7.01}x+\frac{2.53}{7.01}=-1.349501x+0.360913$

3. 3

   Collect the left-hand side.

   $19.2192x+0.208768x-51.2688+1.695198=19.427968x-49.573602$

4. 4

   Bring all x terms to the left and all constants to the right.

   $19.427968x+1.349501x=0.360913+49.573602$

   $20.777469x=49.934515$

5. 5

   Divide and round.

   $x=\frac{49.934515}{20.777469}=2.403301\approx 2.40$

   $\boxed{x=2.40}$

6. 6

   Check by substituting the unrounded value. With $x=2.403301$: the left side is $5.28(8.7480-9.71)+\tfrac{10.5233}{4.79}=-5.0793+2.1969=-2.8823$, and the right side is $-\tfrac{22.7352-2.53}{7.01}=-\tfrac{20.2052}{7.01}=-2.8823$. The two sides agree to four decimal places, confirming the solution.