Roderick takes one look at the equation below and says there are no solutions. |x − 8| + 3 = −4Is he right, and what is his reasoning?A.Yes — absolute value equations cannot have negative numbers.B.Yes — the sum of an absolute value expression and a positive number cannot be negative.C.No — absolute value equations can have negative numbers.D.No — Roderick did not check for extraneous solutions.

Roderick is correct in saying that there are no solutions. The correct reasoning is:

**B. Yes — the sum of an absolute value expression and a positive number cannot be negative.**

Here’s why:

1. The absolute value of any number, \( |x - 8| \), is always **non-negative** (it is always \( \geq 0 \)).
2. Adding 3 to any non-negative number will result in a number that is **greater than or equal to 3**.
3. Therefore, \( |x - 8| + 3 \geq 3 \), and this sum can **never equal -4**.
   
Thus, it is impossible for the equation \( |x − 8| + 3 = -4 \) to have any solutions, because the left side will always be 3 or larger, while the right side is a negative number.

Would you like more details or have any questions?

Here are 5 related questions:

1. What are the general steps to solve absolute value equations?
2. Can the equation \( |x - 8| = -4 \) ever have solutions?
3. How do you check for extraneous solutions in absolute value equations?
4. What is the definition of absolute value in mathematics?
5. Can an absolute value expression ever equal a negative number?

**Tip:** Always check whether the equation has a feasible range of values before solving. For absolute values, they cannot be negative!

Roderick takes one look at the equation below and says there are no solutions. |x − 8| + 3 = −4. Is he right, and what is his reasoning? A. Yes — absolute value equations cannot have negative numbers. B. Yes — the sum of an absolute value expression and a positive number cannot be negative. C. No — absolute value equations can have negative numbers. D. No — Roderick did not check for extraneous solutions.

Understand why the equation |x − 8| + 3 = −4 has no solutions. Explore the absolute value theorem, which states that absolute value expressions are non-negative, and why adding a positive number to it makes the equation impossible to solve. This problem is suitable for Grades 7-10.

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