Question

How do you solve $ \left| {2x + 5} \right| = 3x + 4 $ and find any extraneous solutions?

Answer 1

Hint: The given equation has a modulus function in it, which puts negative or positive signs according to its argument. Split the problem into two cases, in first case open the modulus function with positive sign and in second case open it with sign and then solve the equation in both cases and finally you will get two answers, check the answers with the domain of the cases you have considered.

Complete step-by-step answer:
To solve the equation $ \left| {2x + 5} \right| = 3x + 4 $ we need to open the modulus function which will result in splitting the solution into two cases due to modulus will open with positive and negative sign according to the argument in it.
For the equation $ \left| {2x + 5} \right| = 3x + 4 $
Let us take the first case in which
 $ 2x + 5 \geqslant 0 \Rightarrow x \geqslant \dfrac{{ - 5}}{2} $
Hence argument of modulus is positive, so opening it with positive sign,
 $
   \Rightarrow \left| {2x + 5} \right| = 3x + 4 \\
   \Rightarrow 2x + 5 = 3x + 4 \;
  $
Now solving it for value of $ x $
 $
   \Rightarrow 2x + 5 = 3x + 4 \\
   \Rightarrow 5 - 4 = 3x - 2x \\
   \Rightarrow 1 = x \;
  $
 $ \because x = 1 \geqslant \dfrac{{ - 5}}{2} $
Therefore $ x = 1 $ is a solution for the equation $ \left| {2x + 5} \right| = 3x + 4 $
Now taking the second case in which
 $ 2x + 5 < 0 \Rightarrow x < \dfrac{{ - 5}}{2} $
Here argument of modulus is negative, so opening it with negative sign,
 $
   \Rightarrow \left| {2x + 5} \right| = 3x + 4 \\
   \Rightarrow - (2x + 5) = 3x + 4 \\
  $
Solving it for value of $ x $
 $
   \Rightarrow - 2x - 5 = 3x + 4 \\
   \Rightarrow - 5 - 4 = 3x + 2x \\
   \Rightarrow 5x = - 9 \\
   \Rightarrow x = \dfrac{{ - 9}}{5} \;
  $
Since the value $ x $ should be less than $ \dfrac{{ - 5}}{2} $ for this case, whereas $ \dfrac{{ - 9}}{5} > \dfrac{{ - 5}}{2} $ due to which $ x = \dfrac{{ - 9}}{5} $ is the extraneous solution of the equation $ \left| {2x + 5} \right| = 3x + 4 $
So, the correct answer is “ $ x =1 $ ”.

Note: Problems including modulus functions can be solved by one more way, which is to square up the both sides of the equation and then solve it. You will get two values (one actual and one extraneous) after solving the equation, substitute them in the equation and check for the actual one. Try this method by yourself and compare the results.