Verified Answer :

  • Divide the original equation \(-2|3x - 11| = 0\) by \(-2\) to get \(|3x - 11| = 0\).
  • Apply the property of absolute values to set the argument \(3x - 11 = 0\).
  • Solve the linear equation for \(x\) to find the single solution \(x = \frac{11}{3}\).
  • Conclude that there is only one real solution, which matches option (b).
  • The final answer is \(\boxed{1}\).

Explanation

  1. Analysis of the Equation

We are tasked with finding the number of real solutions for the equation \(-2|3x - 11| = 0\). An absolute value expression, denoted as \(|A|\), represents the non-negative distance of a value from zero on the number line. Specifically, \(|A| = 0\) if and only if the expression inside the absolute value, \(A\), is equal to zero. This property will guide us in solving the given equation.

  1. Isolating and Simplifying the Absolute Value

To isolate the absolute value expression, we divide both sides of the equation by \(-2\):
\[-2|3x - 11| = 0\]
\[|3x - 11| = \frac{0}{-2}\]
\[|3x - 11| = 0\]
By the property of the absolute value, this equation is satisfied if and only if the expression inside the absolute value is equal to zero:
\[3x - 11 = 0\]

  1. Solving for x and Conclusion

Now we solve the linear equation \(3x - 11 = 0\) for \(x\):
\[3x = 11\]
\[x = \frac{11}{3}\]
Since there is exactly one value of \(x\) that satisfies the equation, the equation has exactly one real solution. Comparing this to the given options, we find that the correct choice is (b).

Examples

Absolute value equations appear frequently in engineering and science when dealing with tolerances or margins of error. For example, if a machine part must have a length of exactly \(11/3\) cm, the condition for its production could be modeled by an equation like \(-2|3x - 11| = 0\). If the measurement deviates from this value, the expression \(-2|3x - 11|\) becomes non-zero, indicating that the part does not meet the strict tolerance. Finding the root of such an expression helps identify the specific operating point or ideal value for manufacturing.

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