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: Here, we will solve for the given variable. We will solve the inequality by using the arithmetic operations and we will get the required answer. The solution for this inequality has to be represented on a number line. An inequality is defined as a non-equal comparison between two numbers or expressions with the variables which may be greater than or lesser than or greater than equal to or less than equal to.

Complete step-by-step answer:
We are given an inequality \[x - 3\left( {2 + x} \right) > 2\left( {3x - 1} \right)\].
By multiplying the terms, we get
\[ \Rightarrow x - 6 - 3x > 6x - 2\]
Taking the like terms on one side of the equation, we get
\[ \Rightarrow x - 6x - 3x > 6 - 2\]
By adding the like terms, we get
\[ \Rightarrow x - 9x > 6 - 2\]
By subtracting the like terms, we get
\[ \Rightarrow - 8x > 4\]
By rewriting the equation, we get
\[ \Rightarrow x > \dfrac{4}{{ - 8}}\]
By dividing the numbers, we get
\[ \Rightarrow x > \dfrac{{ - 1}}{2}\]
\[ \Rightarrow x > - 0.5\]
We are given that \[x \in \left| { - 3, - 2, - 1,0,1,2} \right|\]
So, we get
\[ \Rightarrow x = - 1, - 2, - 3\] since \[x > - 0.5\]
Now we will represent the solution on the number line.

Therefore, the values of \[x\] are \[ - 3, - 2, - 1\].

Note: Here, we have added the like terms of the given inequality. We should remember that adding the same quantity on both the sides of an inequality does not change its direction. When we multiply or divide the positive number there is no change in the direction. When we multiply or divide by a negative number, there is a change in the direction of an inequality. We have also used the distributive property of multiplication to simplify the terms. Distribution property states that if a number is multiplied to the sum of two numbers then it is given by \[a \cdot \left( {b + c} \right) = ab + ac\].