Question

Solve: \[30-4\left( 2x-1 \right)<30\], where x is a positive integer and graph the solution on the number line.

Answer 1

- Hint: First solve the inequation by doing basic mathematical operations like addition, subtraction, multiplication so that we will get the single variable in x with some inequality. From that we can get the range of x and then plot the solution on the number line.

Complete step-by-step solution -

Given that \[30-4\left( 2x-1 \right)<30\] where x is a positive integer
$\Rightarrow$ \[30-4\left( 2x-1 \right)<30\]
$\Rightarrow$ \[30-4\left( 2x-1 \right)-30<0\]. . . . . . . . . . . . . . . . . (1)
$\Rightarrow$ \[-4\left( 2x-1 \right)<0\]. . . . . . . . . . . . . . . . . . . . . . .. . (2)
$\Rightarrow$ \[4\left( 2x-1 \right)>0\]. . . . . . . . . . . . . . . . . . . . . . . . . .(3)
$\Rightarrow$ \[\left( 2x-1 \right)>0\]. . . . . . . . . . . . . . . . . . . . . . . . . . . .(4)
$\Rightarrow$ \[x>\dfrac{1}{2}\]. . . . . . . . . . . . . . . . . . . . . (5)
\[x\in \left( \dfrac{1}{2},\infty \right)\]

So, the obtained range of the x is \[x\in \left( \dfrac{1}{2},\infty \right)\]

Note: When we swap the right hand side and left hand side, we must also change the direction of the inequality. When we multiply or divide with a negative number we must reverse the inequality. Number line is a straight line with a ‘zero’ point in the middle with positive and negative numbers marked on either side of zero. Here we can plot fractions so we can plot the solution of x in the given range.