Question

How do you solve the linear inequality $-.25+1.75x< -1.75+2.25x$?

Answer 1

To solve the given inequality, we are going to write the constant terms on the one side of the inequality and terms with variable x on the other side of the inequality. Then add or subtract whatsoever is needed on both the sides of the inequality and at last we will get some range of values of x.

Complete step-by-step solution:
The inequality given in the above problem which we have to solve is as follows:
$-.25+1.75x<-1.75+2.25x$
Adding 1.75 on both the sides in the above inequality we get,
$\begin{aligned} & \Rightarrow -.25+1.75+1.75x<2.25x \\\ & \Rightarrow 1.5+1.75x<2.25x \\\ \end{aligned}$
Subtracting 1.75x on both the sides of the above inequality we get,
$\begin{aligned} & \Rightarrow 1.5<2.25x-1.75x \\\ & \Rightarrow 1.5<0.5x \\\ \end{aligned}$
Dividing 0.5 on both the sides we get,
$\begin{aligned} & \Rightarrow \dfrac{1.5}{0.5}< x \\\ & \Rightarrow 3< x \\\ \end{aligned}$
The solution of x in the above inequality is that x will take all values which are greater than 3. **Hence, the solution of x in the above problem in the interval form is as follows:
$x\in \left( -\infty ,3 \right)$ **

Note: In the above solution, we have written the interval in which x will take values. As you can see that the bracket which we have used to close 3 is an open bracket. Open bracket means we are not including that value means we are not including the value 3 in the given solution.
In some questions we will use the following kind of bracket:
$(-\infty ,4]$
Now, the bracket which we have used to close the number “4” is a closed bracket. This means that we are including the number “4”.
So, we have also discussed closed kinds of brackets also. But in this problem, the brackets which we have used are open brackets.