Question

How do you solve $3n+7-7n<-5$ ?

Answer 1

Hint: To solve these types of inequality expressions, we simply have to solve the expression for the unknown variable keeping the inequality in mind. The inequalities can be solved by adding or subtracting variables or constants on both the sides of the inequality rather than just transposing them.

Complete step by step answer:
We have been given an inequality: $3n+7-7n<-5$.
If it were an equation instead of an inequality, we would have simply transposed the variables to one side of the equation and solved it to get our answer. But that would be inappropriate for an inequality-based equation.
So, we will have to add or subtract the variables and constants on both the sides to cancel the terms in the inequality $3n+7-7n<-5$
First of all we will combine all the variable terms that are on one side of the inequality to get,
$\Rightarrow -4n+7<-5$
Now, we will subtract $7$ from both the sides of the inequality to get,
$\Rightarrow -4n+7-7<-5-7$
We will then simplify the above expression to get,
$\Rightarrow -4n<-12$
Now, since the variable has a negative sign with it, we will multiply $\left( -1 \right)$ on both the sides of the inequality to get,
$\Rightarrow -4n\times \left( -1 \right)>-12\times \left( -1 \right)$
Also, the inequality sign will get reversed after the multiplication of $\left( -1 \right)$ on both the sides. Simplifying the above expression, we get
$\Rightarrow 4n>12$
Now, we will divide both the sides of the inequality by $4$ to get,
$\Rightarrow \dfrac{4n}{4}>\dfrac{12}{4}$
Cancelling the numerator and denominator on both the sides of the inequality with their common factor, we get
$\Rightarrow n>3$ , that is, $n\in \left( 3,\infty \right)$

Hence, on solving the given inequality $3n+7-7n<-5$ we get, $n>3$ or $n\in \left( 3,\infty \right)$

Note: The most common mistake that students make while solving these questions is that they generally just transpose all the variables and constants across the inequality and treat it as an equation. This can lead to miscalculations and ultimately a wrong answer.