To solve the equation \( 10|n| - 3 = 37 \), we first isolate the term involving the absolute value.
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Add 3 to both sides: \[ 10|n| - 3 + 3 = 37 + 3 \] \[ 10|n| = 40 \]
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Divide both sides by 10: \[ |n| = \frac{40}{10} \] \[ |n| = 4 \]
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Now we consider the two cases for the absolute value:
Case 1: \( n = 4 \)
Case 2: \( n = -4 \)
Thus, the solutions to the equation are: \[ n = 4 \quad \text{or} \quad n = -4 \]
Final answer: \( n = 4 \) and \( n = -4 \) (the equation has two solutions).