Question

Which equation is an identity?O 3(x - 1) = x + 2(x + 1) + 1Ox-4(x + 1) = -3(x + 1) + 1O 2x + 3 = 1 (4x + 2) + 2(6x - 3) = 3(x + 1) – x-2

Answer 1

Identity equations are always true, no matter the values that the variables take.

We have to calculate for each one, and if the result gives a true statement, then the equation is an identity:

1) 3(x - 1) = x + 2(x + 1) + 1

`\begin{gathered} 3\left(x-1\right)=x+2\left(x+1\right)+1 \\ 3x-3=x+2x+2+1 \\ 3x-3=3x+3 \\ 3x-3x=3+3 \\ 0=6 \end{gathered}`

This is FALSE (for any value of x), so the equation is not an identity.

2) x-4(x + 1) = -3(x + 1) + 1

`\begin{gathered} x-4\left(x+1\right)=-3\left(x+1\right)+1 \\ x-4x-4=-3x-3+1 \\ x(1-4+3)=-2+4 \\ 0=2 \end{gathered}`

This is FALSE, so the equation is not an identity.

3) 2x + 3 = 1 (4x + 2) + 2

`\begin{gathered} 2x+3=14x+2+2 \\ 3-2-2=14x-2x \\ -1=12x \\ x=(-1)/(12) \end{gathered}`

This equation holds true only for x=-1/12, so it is not an identity.

4) (6x - 3) = 3(x + 1) – x-2



`\begin{gathered} \left(6x-3\right)=3\left(x+1\right)-x-2 \\ 6x-3=3x+3-x-2 \\ 6x-3=2x+1 \\ 6x-2x=1+3 \\ 4x=4 \\ x=1 \end{gathered}`

This equation holds true only for x=1, so it is not an identity.

Neither of the options is an identity.