A student solved the following equation using the following steps:

Question

2(2x+2)=4(x+4)
 
4x+4=4x+4
 
4=4

GPT-4o mini · Answer

The student solved the equation $ 2(2x + 2) = 4(x + 4) $ as follows: 1. **Distribute the terms on both sides of the equation**: - Left side: $ 2(2x + 2) = 2 \cdot 2x + 2 \cdot 2 = 4x + 4 $ - Right side: $ 4(x + 4) = 4 \cdot x + 4 \cdot 4 = 4x + 16 $ So, the equation becomes: $$ 4x + 4 = 4x + 16 $$ 2. **Subtract $ 4x $ from both sides**: $$ 4 = 16 $$ This step indicates that the equation $ 4 = 16 $ is false, which tells us that there is no solution to the original equation. The student correctly simplified the equation initially but misinterpreted the result. The conclusion is that the equation $ 2(2x + 2) = 4(x + 4) $ has **no solution** since it reduces to a false statement. So, you can explain to the student that while both sides of the equation say something about $ 4 $, the ultimate conclusion is that they represent inequality, leading to the conclusion that there are no values of $ x $ that satisfy the original equation.

find · Answer

A student solved the following equation using the following steps 2(2x+2)=4(x+4) 4x+4=4x+4 4 = 4 Infinitely many solutions

GPT-4o mini · Answer

The student's work on the equation $ 2(2x + 2) = 4(x + 4) $ leads to an important realization. Let's break down the steps clearly: 1. **Start with the initial equation**: $$ 2(2x + 2) = 4(x + 4) $$ 2. **Distribute on both sides**: - Left side: $ 2(2x + 2) = 4x + 4 $ - Right side: $ 4(x + 4) = 4x + 16 $ So, the equation simplifies to: $$ 4x + 4 = 4x + 16 $$ 3. **Subtract $ 4x $ from both sides**: $$ 4 = 16 $$ This is a contradiction because $ 4 \neq 16 $. ### Conclusion Based on the Steps: - The equation $ 4 = 16 $ is false, which implies that **there are no solutions** to the equation. ### Important Clarification: The statement "Infinitely many solutions" cannot be correct in this case. That statement would apply if, after performing operations, you ended up with a true statement involving the same expressions on both sides (like $ 4x + 4 = 4x + 4 $, and then simplifying leads to $ 0 = 0 $). However, in this instance, we arrived at a false statement, indicating that there are no values of $ x $ that satisfy the original equation. Thus, the correct assertion is that the equation has **no solutions**, not infinitely many.