Question

How do you solve \[|3x - 3|{\text{ }} < - 15\] ?

Answer 1

Hint: Here we are given a less absolute inequality pattern. The problem suggests that there exists a value of \[x\] that can make the statement true. Well, we know, the absolute value of any term is either zero or always positive which can never be smaller than a negative number. So, using this concept we can say that no solutions exist for the given inequality.

Complete step by step answer:
We are given the inequality
\[|3x - 3|{\text{ }} < - 15\]
First of all, let us consider the given equation as \[\left( 1 \right)\]
Therefore, we have
\[|3x - 3|{\text{ }} < - 15{\text{ }} - - - \left( 1 \right)\]
Now in the question we are asked to solve the given inequality.Now we know that the absolute value of any term is either zero or always positive. So, it can not be smaller than any negative number. Since, \[|3x - 3|\] is always positive and \[ - 15\] is negative which implies that \[|3x - 3|\] is always greater than \[ - 15\]. Therefore, the inequality can never be true.

Hence, no solution exists for the given inequality.

Note: Let’s take some test values to verify the above result.
Example-1) If \[x\] is positive, say \[x = 4\]
Therefore, on substituting we get
\[|3\left( 4 \right) - 3| < - 15\]
On solving, we get
\[ \Rightarrow |9|{\text{ }} < - 15\]
We know \[|a| = a\]
Therefore, we have \[9 < - 15\] which is not possible.

Example-2) If \[x\] is zero
Therefore, on substituting we get
\[|3\left( 0 \right) - 3| < - 15\]
On solving, we get
\[ \Rightarrow | - 3|{\text{ }} < - 15\]
We know \[| - a| = a\]
Therefore, we have \[ - 3 < - 15\] which is not possible.

Example-1) If \[x\] is negative, say \[x = - 3\]
Therefore, on substituting we get
\[|3\left( { - 3} \right) - 3| < - 15\]
On solving, we get
\[ \Rightarrow | - 12|{\text{ }} < - 15\]
We know \[| - a| = a\]
Therefore, we have \[ - 12 < - 15\] which is not possible. Hence, in all the three cases, we get the value which is not possible which means there does not exist any value of \[x\] which can satisfy the given inequality. Hence, no solution exists for this inequality.