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Solve the absolute value equation. \[ |5 n+9|+12=5 \] Select the correct choice below and. if necessarv. fill

Ask by Young Hampton. in the United States
Jan 20,2025

Real Tutor Solution

Tutor-Verified Answer

Answer

There is no solution to the equation.

Solution

To solve the absolute value equation: \[ |5n + 9| + 12 = 5 \] **Step 1: Isolate the Absolute Value** Subtract 12 from both sides of the equation: \[ |5n + 9| = 5 - 12 \] \[ |5n + 9| = -7 \] **Step 2: Analyze the Absolute Value** The absolute value of any real number is always **non-negative** (i.e., it is either positive or zero). Therefore, an absolute value cannot be equal to a negative number. \[ |5n + 9| = -7 \quad \text{(This is impossible)} \] **Conclusion:** There is **no solution** to the equation because the absolute value cannot be negative. **Answer:** **No solution.**

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Beyond the Answer

To solve the equation \( |5n + 9| + 12 = 5 \), we first isolate the absolute value term. Subtracting 12 from both sides gives us: \[ |5n + 9| = 5 - 12 \] This simplifies to: \[ |5n + 9| = -7 \] Since absolute values cannot be negative, this equation has no solutions. To double-check, remember that the absolute value represents the distance from zero, and distances cannot be negative. Thus, if you ever encounter a negative value after isolating the absolute value, you can safely conclude that there are no solutions.

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