Describe what a solution of “no solution” means when solving an absolute value equation.  When and why does this happen?

When solving an absolute value equation, the phrase "no solution" means that there is no value for the variable that can satisfy the equation. This situation arises when the equation involves an absolute value expression set equal to a negative number.

### Why Does This Happen?
The absolute value of a number represents its distance from zero on the number line, which is always non-negative (either positive or zero). Therefore, the absolute value of any expression cannot be negative.

### When Does This Happen?
"No solution" occurs specifically when an equation has the form:

\[
|A| = B
\]

Where \( A \) is an expression involving the variable, and \( B \) is a negative number. Since the absolute value \( |A| \) can never equal a negative number, there is no possible value of the variable that can satisfy the equation.

### Example:
Consider the equation:

\[
|x + 3| = -5
\]

The absolute value \( |x + 3| \) represents a non-negative number, but the equation states it equals \(-5\), a negative number. Since no value of \( x \) can make the absolute value of \( x + 3 \) equal \(-5\), the equation has no solution.

Would you like more details or have any questions?

Here are some related questions:
1. How do you solve absolute value equations when there is a solution?
2. What are some examples of absolute value equations with multiple solutions?
3. How does an absolute value inequality differ from an absolute value equation?
4. Can absolute value equations have an infinite number of solutions?
5. How do you graph absolute value equations on a coordinate plane?

**Tip:** Always check if the absolute value is set equal to a negative number before solving, as it can immediately indicate that there’s no solution.

Understand the concept of “no solution” in absolute value equations. Explore why this situation occurs and when to identify it in equations involving absolute values. Suitable for Grades 9-12.

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