Solve Equation for x: Step-by-Step Calculation

The problem requires finding the value of $x$ in the given equation:

$8.5 - \left\{ 5\frac{1}{2} - \left( 7\frac{1}{2} + 2.8 \div x \right) \right\} \times 4.25 \div (0.2)^2 = 306$

Equation Simplification

First, convert mixed fractions and simplify the power term:

• $5\frac{1}{2} = 5.5$
• $7\frac{1}{2} = 7.5$
• $(0.2)^2 = 0.04$

Substitute these values back into the equation:

$8.5 - \left\{ 5.5 - \left( 7.5 + \frac{2.8}{x} \right) \right\} \times 4.25 \div 0.04 = 306$

Isolating the Variable x

The goal is to isolate $x$ by performing inverse operations step-by-step:

1. Subtract 8.5 from both sides: $-\left\{ 5.5 - \left( 7.5 + \frac{2.8}{x} \right) \right\} \times 4.25 \div 0.04 = 306 - 8.5$ $-\left\{ 5.5 - \left( 7.5 + \frac{2.8}{x} \right) \right\} \times 4.25 \div 0.04 = 297.5$
2. Multiply both sides by 0.04: $-\left\{ 5.5 - \left( 7.5 + \frac{2.8}{x} \right) \right\} \times 4.25 = 297.5 \times 0.04$ $-\left\{ 5.5 - \left( 7.5 + \frac{2.8}{x} \right) \right\} \times 4.25 = 11.9$
3. Divide both sides by 4.25: $-\left\{ 5.5 - \left( 7.5 + \frac{2.8}{x} \right) \right\} = \frac{11.9}{4.25}$ $-\left\{ 5.5 - \left( 7.5 + \frac{2.8}{x} \right) \right\} = 2.8$
4. Multiply both sides by -1: $5.5 - \left( 7.5 + \frac{2.8}{x} \right) = -2.8$
5. Isolate the term containing $x$: $-\left( 7.5 + \frac{2.8}{x} \right) = -2.8 - 5.5$ $-\left( 7.5 + \frac{2.8}{x} \right) = -8.3$
6. Multiply both sides by -1: $7.5 + \frac{2.8}{x} = 8.3$
7. Isolate the fraction term: $\frac{2.8}{x} = 8.3 - 7.5$ $\frac{2.8}{x} = 0.8$
8. Solve for $x$: $x = \frac{2.8}{0.8}$ $x = \frac{28}{8}$ $x = 3.5$

Final Answer

The value of $x$ that satisfies the equation is 3.5.