Question

Solve the following inequation and write down the solution set:
\[11x - 4 < 15x + 4 \le 13x + 14,x \in {\bf{W}}\]

Answer 1

Hint:
Here, we will solve the first inequality and then the next inequality by using basic mathematical operations. Then we will find the solution set of the equation by combining both the inequalities obtained after solving. Inequality is defined as a non-equal comparison between two numbers and two mathematical 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 solution:
We are given with an inequality \[11x - 4 < 15x + 4 \le 13x + 14,x \in {\bf{W}}\]
Now, let us consider the given first two inequalities.
\[11x - 4 < 15x + 4\]
Taking the variables on one side and the constant term, we get
\[ \Rightarrow - 4 - 4 < 15x - 11x\]
Adding the constant term and the variables on both the sides, we get
\[ \Rightarrow - 8 < 4x\]
Dividing by \[ - 4\] on both the sides, we get
\[ \Rightarrow \dfrac{{ - 8}}{4} < \dfrac{{4x}}{4}\]
\[ \Rightarrow - 2 < x\]………………………………………………………………………………………….\[\left( 1 \right)\]
Now, let us consider the given next two inequalities.
\[15x + 4 \le 13x + 14\]
Taking the variables on one side and the constant term, we get
\[ \Rightarrow 15x - 13x \le 14 - 4\]
Adding the constant term and the variables on both the sides, we get
\[ \Rightarrow 2x \le 10\]
Dividing by \[2\] on both the sides, we get
\[ \Rightarrow \dfrac{{2x}}{2} \le \dfrac{{10}}{2}\]
\[ \Rightarrow x \le 5\]………………………………………………………………………………………….\[\left( 2 \right)\]
Combining equations \[\left( 1 \right)\] and \[\left( 2 \right)\] , we get
\[ - 2 < x \le 5\]

Therefore, the solution set for the inequation \[11x - 4 < 15x + 4 \le 13x + 14,x \in {\bf{W}}\] is \[ - 2 < x \le 5\] i.e., \[x = 0, 1, 2, 3, 4, 5\]

Note:
Here, 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.