A machine prints trading cards and then trims them to the correct size. The absolute value equation |w - 3.5|= 0.003 gives the width, in inches, allowed for the cards to be certified authentic. What are the allowed widths, w, according to the equation?

Answer 1

To solve the absolute value equation \( |w - 3.5| = 0.003 \), we need to consider the definition of absolute value. The equation \( |x| = a \) implies that \( x = a \) or \( x = -a \).

In our case, we set \( w - 3.5 = 0.003 \) and \( w - 3.5 = -0.003 \):

1. For the first case: \[ w - 3.5 = 0.003 \] Adding \( 3.5 \) to both sides gives: \[ w = 3.5 + 0.003 = 3.503 \]

2. For the second case: \[ w - 3.5 = -0.003 \] Adding \( 3.5 \) to both sides gives: \[ w = 3.5 - 0.003 = 3.497 \]

Thus, the allowed widths \( w \) that satisfy the equation are: \[ w = 3.503 \text{ inches and } w = 3.497 \text{ inches.} \]