Thinking Mathematically (6th Edition)

In solving an equation with variable x, if you eliminate the variable and obtain a statement such as$6=6$, the equation is true for every value of x. The solution set can be expressed in set builder notation as, $\left\{ x|x\text{ is a real number} \right\}$.
While solving an equation if the variable on both the sides of an equation gets eliminated and still left-hand side equals the right-hand side, then the equation is correct or true for every value of x, because no matter what value x takes, the equation is satisfied by its own. The representation for the same is given as$\left\{ x|x\text{ is a real number} \right\}$. This means for every real value of x, it will satisfy the equation.