The expressions \( 4(x+1)-1 \) ond \( 2 x+2(x+1) \) are not equivalene. Explain how the distributive property and the commutative property of addition are used in the Example to show that \( 2(x+1)+2 x \) is equivalent to \( 4 x+2 \).

To show that \(2(x+1) + 2x\) is equivalent to \(4x + 2\), we start by applying the distributive property to \(2(x+1)\). When we distribute, we get \(2 \cdot x + 2 \cdot 1\), resulting in \(2x + 2\). Now, when we add \(2x\) to this, we have \(2x + 2 + 2x\). Here is where the commutative property comes into play: we can rearrange our terms to group the \(2x\)s together. So we combine like terms, yielding \(2x + 2x + 2 = 4x + 2\). Isn't math like a magic show? You know, you have one expression that looks all different, and with the magical powers of properties like distributive and commutative, you reveal an equivalent one! The rules help you transform equations, making it possible to see the hidden connections. Once you master these tricks, you can tackle even trickier math with confidence!