Question

at a pumpkin patch, if armando guesses the weight of his pumpkin within 0.3 pounds, he gets to take the pu home for free. if his pumpkin weighs 4.9 pounds, which two equations can be used to find the minimum and maximum weights he can guess in order to get his pumpkin for free?
\\( x - 0.3 = 4.9 \\) and \\( x - 0.3 = -4.9 \\)
\\( x + 0.3 = 4.9 \\) and \\( x + 0.3 = -4.9 \\)
\\( x + 4.9 = 0.3 \\) and \\( x + 4.9 = -0.3 \\)
\\( x - 4.9 = 0.3 \\) and \\( x - 4.9 = -0.3 \\)

Explanation:

Step1: Understand the problem

Let \( x \) be the guessed weight. The actual weight is \( 4.9 \) pounds, and the difference between the guessed weight and the actual weight should be within \( 0.3 \) pounds. This means \( |x - 4.9| = 0.3 \).

Step2: Rewrite the absolute - value equation

The absolute - value equation \( |a|=b\) (where \( b\geq0\)) can be rewritten as \( a = b\) or \( a=-b\). For \( |x - 4.9| = 0.3\), we can rewrite it as \( x - 4.9 = 0.3\) (this will give the maximum guessed weight, since when we solve for \( x\), we add \( 4.9\) to both sides: \( x=4.9 + 0.3\)) and \( x - 4.9=- 0.3\) (this will give the minimum guessed weight, since when we solve for \( x\), we add \( 4.9\) to both sides: \( x = 4.9-0.3\)).

Let's analyze the other options:

• Option 1: \( x - 0.3 = 4.9\) and \( x - 0.3=-4.9\) is incorrect. The difference should be between \( x\) and \( 4.9\), not between \( x\) and \( 0.3\).
• Option 2: \( x + 0.3 = 4.9\) and \( x + 0.3=-4.9\) is incorrect. The relationship is about the difference between the guessed weight and the actual weight (\( 4.9\)), not about \( x+0.3\) and \( 4.9\) or \(- 4.9\).
• Option 3: \( x + 4.9 = 0.3\) and \( x + 4.9=-0.3\) is incorrect. The equation is not set up correctly. The correct relationship is about the difference between \( x\) and \( 4.9\), not between \( x\) and \(-4.9\) with respect to \( 0.3\).

Answer:

\( x - 4.9 = 0.3\) and \( x - 4.9=-0.3\) (the last option: \( x - 4.9 = 0.3\) and \( x - 4.9=-0.3\))