Question

How do you solve the inequality is \[2n + 5 > 11 - n?\]

Answer 1

Hint: In this question we solve single linear inequalities and follow pretty much the same process for solving linear equations. We will simplify both sides, get all the terms with the variable on one side and the numbers on the other side, and then multiply or divide both sides by the coefficient of the variable to get the solution.

Complete step-by-step solution:
To find the inequality of $n$ from the given linear inequality:
Given:
\[ \Rightarrow 2n + 5 > 11 - n\]
First, we rearrange the equation with same sign to the right hand side (RHS) only of the above equation and we get
\[ \Rightarrow 2n + 5 > - n + 11\]
Next, we adding the $( - 5 + n)$ in both side of the above equation and we get
$ \Rightarrow 2n + 5 + ( - 5 + n) > - n + 11 + ( - 5 + n)$
Now, we replace the term in parentheses on both sides and solve the above equation and we get
$ \Rightarrow (2n + 5) + (n - 5) > ( - n + 11) + (n - 5)$
Next, we remove the parentheses and solve the above equation and we get
$ \Rightarrow 2n + 5 + n - 5 > - n + 11 + n - 5$
Then, we rearrange the variable and numbers on both sides and we get
$ \Rightarrow 2n + n + 5 - 5 > - n + n + 11 - 5$
Next, subtracting the number only on both sides and we get
$ \Rightarrow 2n + n > - n + n + 6$
Next, adding the variable in left hand side (LHS) and subtracting the variable in right hand side (RHS) and we get
$ \Rightarrow 3n > 6$
Now, we divide both sides by 3 and we get
$ \Rightarrow n > 2$ is the solution.

The interval notation is $n \in (2,\infty )$

Note: we have to mind that, a linear inequality looks exactly like a linear equation, with the inequality sign replacing the equality sign. Linear inequalities are the expressions where any two values compared by the inequality symbols such as $' < ',' > ',' \leqslant 'or' \geqslant '.$
You can get inequalities that are false for any variable. These inequalities have no solution. You can also get inequalities that are true for any variable. These have an infinite number of solutions.
There is another little hard way to find the answer.
First, treat it like it’s an equal sign, and solve like you would algebra.
So,
\[ \Rightarrow 2n + 5 > 11 - n\]
Next, move all terms not containing $n$ to the right hand side of the inequality (RHS) and move all terms containing $n$ to the left hand side (LHS). Also, signs are change on both sides and we get
\[ \Rightarrow 2n + n > 11 - 5\]
Next, simply on both sides and we get
$ \Rightarrow 3n > 6$
Then, the number 3 move in right hand side (RHS) and we get
$ \Rightarrow n > \dfrac{6}{3}$
Next, we divide in right hand side (RHS) of the above equation and we get the final answer:
$ \Rightarrow n > 2$
Therefore, the solution is $n > 2.$