Question

How do you solve the equation \[\left| 6-2x \right|=6\]?

Answer 1

Hint: In this problem, we have to solve and find the value of x. Here we are given an absolute value equation. We know that an absolute value equation will always be positive. We have to use the property of absolute value equations. We can first apply the rule if \[\left| x \right|=k\] then \[x=k\] or \[x=-k\]. We can then solve the resulting equations, and put the solution into the original equation to get the correct root.

Complete step by step answer:
We know that the given absolute equation to be solved is,
 \[\Rightarrow \left| 6-2x \right|=6\]……. (1)
We are given an absolute value equation. We know that an absolute value equation will always be positive. We have to use the property of absolute value equations.
We can first apply the rule if \[\left| x \right|=k\] then \[x=k\] or \[x=-k\].
In this problem, if \[\left| 6-2x \right|=6\] then \[6-2x=6\] or \[6-2x=-6\], we can apply this rule in the given equation and solve it.
Case1:
If \[6-2x=6\], then solve for x,
\[\begin{align}
  & \Rightarrow -2x=0 \\
 & \Rightarrow x=0 \\
\end{align}\]
We can now substitute this value in (1), we get
\[\begin{align}
  & \Rightarrow \left| 6-2\left( 0 \right) \right|=6 \\
 & \Rightarrow \left| 6 \right|=6 \\
\end{align}\]
We can see that the case1 satisfies the condition.
Case2:
If \[6-2x=-6\], then solve for x,
\[\begin{align}
  & \Rightarrow 6+6=2x \\
 & \Rightarrow x=6 \\
\end{align}\]

Therefore, the value of x = 0, 6.

Note: Students make mistakes while following the steps to solve the absolute value equations. We should first apply the rule and solve for x. We should then substitute the value of x, to check whether the condition for the property of absolute value, i.e. the rule if \[\left| x \right|=k\] then \[x=k\] or \[x=-k\] is correct.