Question

Solve \[x - 3\left( {2 + x} \right) > 2\left( {3x - 1} \right)\] , \[x \in \left\{ { - 3, - 2, - 1,0,1,2} \right\}\]. Also represent its solution on the number line.

Answer 1

Hint: In this question, linear inequation is given to us. Also it is given to us that \[x \in \left\{ { - 3, - 2, - 1,0,1,2} \right\}\] . So we have to first solve the linear inequation and find the solution set with the help of the values given to us to which x belongs to. Then mark that solution set on the number line by putting a dot to simply highlight the solution set.

Complete step by step answer:
A solution set is the set of values which satisfy a given inequality. It means, each and every value in the solution set will satisfy the inequality and no other value will satisfy the inequality. The solutions of an inequality can be represented on a number line.
A statement indicating that the value of one quantity or algebraic expression which is not equal to another is called an inequation.
An inequation which involves only one variable whose highest power one is known as a linear inequation in that variable.
So, the given linear inequation is
\[ \Rightarrow x - 3\left( {2 + x} \right) > 2\left( {3x - 1} \right)\]
By opening the brackets on both sides we get
\[ \Rightarrow x - 6 - 3x > 6x - 2\]
On subtracting the like terms in the left side we get
\[ \Rightarrow - 6 - 2x > 6x - 2\]
On shifting \[ - 2x\] to the right hand side and \[ - 2\] to the left hand side we get
\[ \Rightarrow - 6 + 2 > 6x + 2x\]
By adding terms on both sides we get
\[ \Rightarrow - 4 > 8x\]
On dividing eight on both sides we have
 \[ \Rightarrow \dfrac{{ - 4}}{8} > x\]
On further solving we get
\[ \Rightarrow \dfrac{{ - 1}}{2} > x\]
Or \[ \Rightarrow x < \dfrac{{ - 1}}{2}\]
Now it is given to us that \[x \in \left\{ { - 3, - 2, - 1,0,1,2} \right\}\] . In this the values of x: \[ - 3\] , \[ - 2\] and \[ - 1\] are less than \[\dfrac{{ - 1}}{2}\] .
Therefore, the solution set of the given linear inequation is \[\left\{ { - 3, - 2, - 1} \right\}\] .
Let us mark the solution set on the number line as shown below.

The solution set is represented on the number line by blue dots.

Note:
Keep in mind that the values of the variables which make the inequality a true statement are called its solutions. Remember that linear inequation looks exactly like a linear equation with an inequality sign replacing the equality sign. Note that we have to use curly brackets for set notation.