How do you solve the equation $7\left| x+3 \right|=21$?

Hint: To solve this equation first we will divide the whole equation by 7. Now we know that $\left| x \right|=x,x>0$ and $|x|=-x,x<0$ . Hence using this we will open the modulus and rewrite the equation. Now we have two cases, first when x + 3 > 0 and second when x + 3 < 0. For both cases we will solve the linear equation and hence find the solution to the given equation.

Complete step by step solution:
Consider the given equation $7\left| x+3 \right|=21$
Now dividing the whole equation by 7 we get $\left| x+3 \right|=3.......\left( 1 \right)$
Now the modulus function is defined as
$\begin{align}
  & \left| x \right|=x,x>0 \\
 & |x|=-x,x<0 \\
\end{align}$
Now using the definition of modulus function we open the given modulus $\left| x+3 \right|$
$\left| x+3 \right|=x+3,x+3>0\Rightarrow \left| x+3 \right|=x+3,x>-3.......\left( 2 \right)$
and $\left| x+3 \right|=-x-3,x+3<0\Rightarrow \left| x+3 \right|=-x-3,x<-3.......\left( 3 \right)$
Now first let us consider the case where x > - 3.
Then from equation (2) we know that $\left| x+3 \right|=x+3$
Hence substituting this in equation (1) we get $x+3=3$
Solving the above equation we get one solution of the equation as x = 0
Now consider the case where x < - 3.
Then from equation (3) we know that $\left| x+3 \right|=-x-3$
Now substituting this equation (1) we get, $-x-3=3$
Hence rearranging the terms we get another solution of the equation as $x=-6$
Hence we get the solution of the equation are x = 0 and x = - 6.

Note: For this problem we can also solve directly. Since we are given $\left| x+3 \right|=3$ then there are just two possibilities either x + 3 = 3 or x + 3 = - 3 as we know that $\left| 3 \right|=\left| -3 \right|=3$. Hence we will solve these linear equations and find the value of x in both cases.