Question

How do you solve $ 4x + 10 = 2(2x + 5) $ ?

Answer 1

Hint: The first step we need to do every time we are solving a linear equation with a single variable is that you will have to open up every bracket possible and isolate the unknown variable out. So, just after opening the brackets, you will notice that upon solving you get the answer as $ 0 = 0 $ . This clearly means that the given equation is an identity and hence valid for all real numbers that exist.

Complete step-by-step solution:
The given question we have is $ 4x + 10 = 2(2x + 5) $
This is clearly an equation with a single variable of unity power. To solve this, all we have to do is isolate the unknown variable out from the rest and try to equate the other things against it. Now, to do it, we will open the brackets on the RHS to get a much simplified version of our equation
 $
  4x + 10 = 2(2x + 5) \\
   = 4x + 10 \\
  $
Subtracting 10 from both LHS and RHS of the equation
 $
  4x + 10 - 10 = 4x + 10 - 10 \\
  4x = 4x \\
 $
Dividing 4 on both RHS and LHS, we will get
 $
  \dfrac{{4x}}{4} = \dfrac{{4x}}{4} \\
  x = x \\
 $
Bringing like terms “x” on the same side of the equation
 $ x - x = 0 \\
  0 = 0 \\
 $
Surprisingly, the terms when solved gave us a value of $ 0 = 0 $ which means only one thing and that is:-
The given equation to us in the question is an identity. And we need to remember that every identity equation where we will get $ 0 = 0 $ in the end Is true for every real number that exists.
Hence, we can safely conclude that the solution of the given question is $ ( - \infty ,\infty ) $

Note: Please note that, for an equation to be an identity, you must end up with equal terms on both the side of the equation. The only $ 0 = 0 $ is considered as an identity condition and not $ nx = 0 $ format where n Is any real number. So refrain from concluding that equation is identity for the second case mentioned here.