How can you determine if an absolute value equation has no solution?

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To determine if an absolute value equation has no solution, we first need to understand the nature of absolute values. An absolute value equation typically looks like this:

 |ax + b| = c

In this equation:

  • ax + b is a linear expression where a and b are constants,
  • c is a non-negative number as absolute values cannot produce negative results.

Here’s a step-by-step method to determine if such an equation has no solution:

  1. Check the nature of c: First, observe the right side of the equation, c. If c is a negative number, the equation has no solution because the absolute value can’t equal a negative value.
  2. Set up the related equations: If c is non-negative, proceed to set up two equations based on the definition of absolute value:
  3.  ax + b = c 
  4.  ax + b = -c
  5. Solve for x: Solve both equations to find potential solutions:
  6. x_1 = (c - b) / a 
  7. x_2 = (-c - b) / a
  8. Check for contradictions: After obtaining the solutions, substitute them back into the original absolute value equation to see if they hold true. If both solutions are invalid in the context of the original equation, or if they yield a contradiction, it indicates there are no solutions.

For example, consider the equation:

 |2x + 3| = -1

Here, since -1 is negative, there are no solutions because an absolute value cannot equal a negative number.

In summary, the key indicators of an absolute value equation having no solution are:

  • The right side of the equation (c) is negative.
  • If valid, both derived equations do not satisfy the original equation.

By following these steps and understanding the implications of absolute values, you can efficiently determine if an absolute value equation has no solution.

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