Question

How do you solve \[|x| > 12\]?

Answer 1

Hint: To solve this question, we will make two cases using the concept that \[|x| = x\], when \[x\] is positive and \[|x| = - x\], when \[x\] is negative. Then, we will calculate the values of \[x\], from each case and plot them on the number line. Then we will choose the appropriate region from the number line according to the question.

Complete step by step answer:
Given;
\[|x| > 12\]
Case 1: \[x\] is positive.
We know when \[x\] is positive then; \[|x| = x\]. So, with this concept we get;
\[ \Rightarrow x > 12\]
Case 2: \[x\] is negative.
We know that when \[x\] is negative then, \[|x| = - x\]. So, using this we get;
\[ \Rightarrow - x > 12\]
Multiplying both sides with \[ - 1\], we get
\[ \Rightarrow x < - 12\]
Here the sign of inequality changed because we multiply inequality by -1.
Now we will plot both cases on the number line.

for the first case \[x > 12\],
\[ \Rightarrow x \in \left( {12,\infty } \right)\]
This is represented by region 1 on the number line.
For the second case \[x < - 12\],
\[ \Rightarrow x \in \left( { - \infty , - 12} \right)\]
This is represented by region 2 on the number line.
Now \[x\] can belong to any of these regions. So, we will take the union of both the regions to represent the range of \[x\].
\[\therefore x \in \left( { - \infty , - 12} \right) \cup \left( {12,\infty } \right)\]

Note:
One thing to note here is that we have not included \[12\] in our result. This is because the equality sign is not given in the question. Suppose if the question would have been \[|x| \geqslant 12\], then we would have included \[12\] in our answer. One more thing to note is the use of brackets. While writing the solution we have used the parentheses. If we use the square braces then our answer will be wrong this is because square braces include the endpoints written inside it and here, we cannot include the endpoints i.e., \[ - \infty , - 12{\text{ or 12,}}\infty \].