Question

Solve the following inequality: - $2x-1<5$

Answer 1

Hint: This is a kind of linear inequality which can be solved simply by taking constants on one side and the terms containing x on the other side and then solving it.

Complete step-by-step answer:
In the question above, we are given a linear inequality. A linear inequality is a kind of inequality which contains only one variable term and constant terms. Thus, an equation of the form $ax+b<0$ is a linear inequality where a and b are any constants. Thus, equation can be solved first taking b on the right side of the inequality i.e.
$ax<-b$
Now, if a is positive, cross multiply ‘a’ and we don’t change sign of inequality i.e.,
$x<\dfrac{-b}{a}$
This means that x should be less than $\dfrac{-b}{a}$. In other terms we can write it as:
$x\in \left( -\infty ,\dfrac{-b}{a} \right)$

But if a is positive, we will cross multiply a and we will change the sign of inequality i.e.
$x>\dfrac{-b}{a}$
This means that x should be greater than $\dfrac{-b}{a}$. In other terms, we can rewrite it as
$x\in \left( \dfrac{-b}{a},\infty \right)$
Now, we will solve the question given as
$2x-1<5$
We have to separate constant terms and terms containing x. thus after separating, we get: -
$\Rightarrow 2x<5+1$
$\Rightarrow 2x<6$
$\Rightarrow x<\dfrac{6}{2}$
$\Rightarrow x<3$
Thus, the final solution of the inequality is $x\in \left( -\infty ,3 \right)$

Note: We can also do this question using graphical methods. In this method, we will draw the line $y=2x-6$. Now, as the question demands $2x-6<0\ \Rightarrow \ y<0$. Thus, the values of x at which y is less than zero will be our required answer. Not only for linear inequality, we can do this for any order of inequality.