Q. Solve Equation 6.9 Divided by x Equals 3/2

Problem

Solve the equation

\[ \frac{6.9}{x} = \frac{3}{2} \]

1. State the domain restriction.

   The expression \( \frac{6.9}{x} \) is undefined when \( x = 0 \), so any solution must satisfy \( x \neq 0 \).

2. Isolate the variable.

   Multiply both sides of the equation by \( x \) to remove the denominator on the left. This yields

   \[ 6.9 = \frac{3}{2}\,x \]

   We may now solve for \( x \) by undoing multiplication by \( \tfrac{3}{2} \).

3. Solve for \( x \) by multiplying by the reciprocal.

   Multiply both sides by the reciprocal of \( \tfrac{3}{2} \), which is \( \tfrac{2}{3} \):

   \[ x = 6.9 \cdot \frac{2}{3} \]

4. Compute the product (exact and decimal forms).

   Write \( 6.9 \) as the fraction \( \tfrac{69}{10} \) and multiply:

   \[ x = \frac{69}{10} \cdot \frac{2}{3} = \frac{138}{30} \]

   Simplify the fraction by dividing numerator and denominator by 6:

   \[ x = \frac{23}{5} \]

   Convert to a decimal if desired:

   \[ x = 4.6 \]

5. Check the solution.

   Substitute \( x = 4.6 \) back into the original equation:

   \[ \frac{6.9}{4.6} = \frac{69/10}{46/10} = \frac{69}{46} = \frac{3}{2} \]

   The left-hand side equals the right-hand side, so the solution is correct and satisfies the domain restriction.

Final answer: \[ x = \frac{23}{5} = 4.6 \]