Question

The set of values of $a$ for which the inequality, $\left( x-3a \right)\left( x-a-3 \right)>0$ is satisfied for all $x\in \left[ 1,3 \right]$
A) $\left( \dfrac{1}{3},3 \right)$
B) $\left( 0,\dfrac{1}{3} \right)$
C) $\left( -2,0 \right)$
D) $\left( -2,3 \right)$

Answer 1

Hint:
Here we need to find the range of the given variable. For that, we will solve the given inequality. We will take three random values of the variable from its given range and we will substitute these values one by one which will form three equations including that variable. From there, we will get the values of the required variable.

Complete step by step solution:
It is given that the value of the variable $x$ lies in the interval $\left[ 1,3 \right]$
We know that the values of $x$ in the interval $\left[ 1,3 \right]$ satisfy the given inequality $\left( x-3a \right)\left( x-a-3 \right)>0$.
Now, we will take only three values of $x$ from the given interval. The three values of are 1,2 and 3.
We will substitute these values one by one in the given inequality.
When $x=1$, inequality becomes;
$\Rightarrow \left( 1-3a \right)\left( 1-a-3 \right)<0$
On further simplification, we get
$\Rightarrow \left( 1-3a \right)\left( a+2 \right)>0$
This is possible when $\left( 1-3a \right)>0$ and $\left( a+2 \right)>0$
Thus, the solutions for above inequality are $a<\dfrac{1}{3}$ and $a>-2$ …….. $\left( 1 \right)$
When $x=2$, inequality becomes;
$\Rightarrow \left( 2-3a \right)\left( 2-a-3 \right)<0$
On further simplification, we get
$\Rightarrow \left( 2-3a \right)\left( a+1 \right)>0$
This is possible when $\left( 2-3a \right)>0$ and $\left( a+1 \right)>0$
Thus, the solutions for above inequality are $a<\dfrac{2}{3}$ and $a>-1$ ……… $\left( 2 \right)$
When $x=3$, inequality becomes;
$\Rightarrow \left( 3-3a \right)\left( 3-a-3 \right)<0$
On further simplification, we get
$\Rightarrow \left( 3-3a \right)\left( a \right)>0$
This is possible when $\left( 3-3a \right)>0$ and $a>0$
Thus, the solutions for above inequality are $a<1$ and $a>0$ ……….. $\left( 3 \right)$
Thus, from equation 1, equation 2 and equation 3, we get
$0
Hence, the correct option is option B.

Note:
We have solved the given inequality to find the value of the variable $a$ . If expressions on the left hand side and right hand sides are unequal, that would be called an inequality equation. We need to know some basic and important properties of inequality. Some important properties of inequality are as follows:-
If we add or subtract a term from both sides of inequality, then inequality remains the same.
If we multiply or divide a positive number on both sides of inequality, then inequality remains the same.
But if we multiply or divide a negative number on both sides of inequality, then inequality does not remain the same.