Which of the following equations has no solution?

bracket[bb](x-7)=5

bracket[bb](x-7)=0

bracket[bb](x-7)=-5

Let's analyze each of the equations to determine which one has no solution.

1. **\( |x - 7| = 5 \)**:
   The absolute value equation \( |x - 7| = 5 \) means the distance between \( x \) and 7 is 5 units. This gives two possible solutions:
   \[
   x - 7 = 5 \quad \text{or} \quad x - 7 = -5
   \]
   Solving these:
   \[
   x = 12 \quad \text{or} \quad x = 2
   \]
   So, this equation has two solutions: \( x = 12 \) and \( x = 2 \).

2. **\( |x - 7| = 0 \)**:
   The absolute value equation \( |x - 7| = 0 \) means the distance between \( x \) and 7 is 0 units. This happens when \( x = 7 \).
   So, this equation has one solution: \( x = 7 \).

3. **\( |x - 7| = -5 \)**:
   The absolute value equation \( |x - 7| = -5 \) is impossible because the absolute value of any expression is always non-negative (i.e., \( |x - 7| \geq 0 \)). Therefore, this equation has **no solution**.

### Conclusion:
The equation **\( |x - 7| = -5 \)** has no solution.

Would you like more details or have any other questions?

Here are five related questions to further explore:
1. What are the properties of absolute value functions?
2. How do you solve general absolute value equations?
3. Can absolute value equations have one solution?
4. What happens when the right-hand side of an absolute value equation is negative?
5. How can you graph absolute value functions?

**Tip**: Always check if the right-hand side of an absolute value equation is non-negative before solving.

Which of the following equations has no solution?

|x - 7| = 5
|x - 7| = 0
|x - 7| = -5

Analyze absolute value equations to determine which one has no solution. This problem involves understanding the properties of absolute values, algebraic manipulation, and equation solving. Suitable for students in Grades 6-8.

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