e.) \( 2^{x+3}-2^{x+2}=2^{x+2} \)

To solve the equation \( 2^{x+3} - 2^{x+2} = 2^{x+2} \), we can start by simplifying the left side. Notice that we can factor out \( 2^{x+2} \) from the left-hand side: \[ 2^{x+2}(2^1 - 1) = 2^{x+2} \Rightarrow 2^{x+2}(2 - 1) = 2^{x+2} \Rightarrow 2^{x+2}(1) = 2^{x+2} \] This means the equation is valid if \( 2^{x+2} \) is not zero. Therefore, we just need to check the validity of this condition itself. Since \( 2^{x+2} \) is always positive for any real number \( x \), our original solution can be simplified to finding the specific \( x \): We can rewrite it cleanly as: \[ 2^{x+3} - 2^{x+2} - 2^{x+2} = 0 \Rightarrow 2^{x+3} - 2 \cdot 2^{x+2} = 0 \Rightarrow 2^{x+3} - 2^{x+3} = 0 \] Thus, the original equation is satisfied for any real number \( x \). So, the solution set is all real numbers \( x \in \mathbb{R} \)!