Question

When you solve the equation x + 1 = 2x for x, the solution is 1. 1) Square both sides of the equation, and verify that your solution satisfies the new equation. 2) Show that −1/3 satisfies the new equation but not the original equation.

Answer 1

Part 1: The solution of the new equation is $x=1$ and $x=-\frac{1}{3}$ and the two solutions satisfies the new equation.

Part 2: The solution $x=-\frac{1}{3}$ satisfies the new equation but not the original equation.

Explanation:

Part 1: The given equation is $x+1=2 x$

We need to square both sides of the equation and verify that the solution satisfies the new equation.

Now, squaring both sides of the equation, we get,

$(x+1)^2=(2 x)^2$

Thus, the new equation is $(x+1)^2=(2 x)^2$

Let us solve the new equation to determine the solution.

         $x^{2} +2x+1=4x^{2}$

$4x^{2}-x^{2} -2x-1=0$

       $3x^{2} -2x-1=0$

Taking factors, we get,

$3x(x-1)+1(x-1)=0$

       $(3x+1)(x-1)=0$

                             $x=-\frac{1}{3} ,1$

Thus, the solution of the new equation is $x=1$ and $x=-\frac{1}{3}$

Now, we shall verify the solution $x=1$ and $x=-\frac{1}{3}$ that satisfies the new equation $(x+1)^2=(2 x)^2$

Substituting $x=1$ in $(x+1)^2=(2 x)^2$, we get,

$(1+1)^2=(2 (1))^2$

     $(2)^2=(2)^2$

         $4=4$

Since, both sides of the equation are equal, the solution $x=1$ satisfies the new equation.

Also, substituting $x=-\frac{1}{3}$ in $(x+1)^2=(2 x)^2$, we get,

$(-\frac{1}{3} +1)^2=(2 (-\frac{1}{3} ))^2$

        $(\frac{2}{3} )^2=(-\frac{2}{3} )^2$

             $\frac{4}{9} =\frac{4}{9}$

Since, both sides of the equation are equal, the solution $x=-\frac{1}{3}$ satisfies the new equation.

Thus, the solutions $x=1$ and $x=-\frac{1}{3}$ satisfies the new equation.

Part 2: Now, we need to show that the solution $x=-\frac{1}{3}$ satisfies the new equation but not the original equation.

Substituting $x=-\frac{1}{3}$ in the new equation, we get,

  $(x+1)^2=(2 x)^2$

$(-\frac{1}{3} +1)^2=(2 (-\frac{1}{3} ))^2$

        $(\frac{2}{3} )^2=(-\frac{2}{3} )^2$

             $\frac{4}{9} =\frac{4}{9}$

Since, both sides of the equation are equal, the solution $x=-\frac{1}{3}$ satisfies the new equation.

Now, substituting $x=-\frac{1}{3}$ in the original equation, we have,

  $x+1=2 x$

$-\frac{1}{3} +1=2 (-\frac{1}{3} )$

        $\frac{2}{3} =-\frac{2}{3}$

Since, both sides of the equation are not equal, the solution $x=-\frac{1}{3}$ does not satisfies the original equation.