Question

How do you solve \[\left| x+4 \right|=10\]?

Answer 1

Hint: The modulus gives the absolute value of its argument. To solve problems of the type \[\left| x \right|=a\], we have to take two cases and solve the two cases separately. The first case is \[x=a\]. And the second case is \[x=-a\]. We will take the same cases for the given modulus problem.

Complete step by step answer:
We are asked to solve \[\left| x+4 \right|=10\]. As we know to solve problems of the type \[\left| x \right|=a\], we have to take two cases and solve the two cases separately. The first case is \[x=a\]. And the second case is \[x=-a\]. Here, we have \[x+4\] at the place of \[x\], and \[a=10\]. We will take the same cases for the given modulus problem.
The first case is, \[x+4=10\]
Subtracting 4 from both sides of the above equation, we get
\[\begin{align}
  & \Rightarrow x+4-4=10-4 \\
 & \therefore x=6 \\
\end{align}\]
One solution value of \[x\] is 6.
The second case is, \[x+4=-10\]
Subtracting 4 from both sides of the above equation, we get
\[\begin{align}
  & \Rightarrow x+4-4=-10-4 \\
 & \therefore x=-14 \\
\end{align}\]
The other solution value of \[x\] is \[-14\].

Thus, the two values satisfying the given modulus are 6, and -14.

Note: It is necessary to know the properties of the modulus to solve its questions. The property of the modulus, we used to solve this problem is \[\left| x \right|=a\Rightarrow x=\pm a\]. There are many other properties of the modulus that we should know some of them are as follows,
If \[\left| x \right| < a\], then \[-a < x < a\]. Given that a is a positive and real.
If \[\left| x \right|>a\], then \[x < -a\] or \[x>a\].
These should be remembered.