Respuesta :
The solution of 6x+8=10x-8 is x=4:
[tex]\begin{gathered} 6x+8=10x-8 \\ 6x-10x=-8-8 \\ -4x=-16 \\ x=4 \end{gathered}[/tex]Part A:
We can write:
[tex]\begin{gathered} (6x+8)+2=(10x-8)+2 \\ 6x+8=10x-8 \\ x=4 \end{gathered}[/tex]The equation in the second step is the same as the first equation, that has the solution set x=4.
Part B:
We have the first equation squared:
[tex]\begin{gathered} (6x+8)^2=(10x-8)^2 \\ \sqrt[]{(6x+8)^2}=\sqrt[]{(10x-8)^2} \\ |6x+8|=|10x-8| \\ 6x+8=10x-8 \\ x=4 \end{gathered}[/tex]Then we have the same equation that we started from.
Part C:
We have a new equation and we have to prove that x=0 is a solution:
[tex]\begin{gathered} (6x+8)^2=(10x-8)^2 \\ (6\cdot0+8)^2=(10\cdot0-8)^2 \\ 8^2=(-8)^2 \\ 64=64\longrightarrow\text{True} \end{gathered}[/tex]Then, x=0 is a solution.
But if we apply the same to the first equation we get:
[tex]\begin{gathered} 6x+8=10x-8 \\ 6\cdot0+8=10\cdot0-8 \\ 8=-8\longrightarrow\text{False (x=0 is not a solution)} \end{gathered}[/tex]Part D:
When both sides of the equation are squared, both sides will have a positive sign. This does not necesarily happen when the sides are not squared.
For example, the equation we have been working for, when squared, is equal to:
[tex]|6x+8|=|10x-8|[/tex]That is why x=0 is a solution for the squared equation but not for the original equation, when both sides end with different sign.
