Question

How do you solve $x\left( x-1 \right)\left( x+2 \right)>0$ ?

Answer 1

Hint: Try to find the values of x by assuming the equation as $x\left( x-1 \right)\left( x+2 \right)=0$ and plot them on the number line. Mark +,–,+,… alternatively between the points from right to left. Then find the range according to the table of Wavy Curve Method.

Complete step-by-step answer:
To find the roots of the equation $x\left( x-1 \right)\left( x+2 \right)>0$ we have to equate each factor to ‘0’ separately.
So the roots are
$x=0$
Or,
$\begin{align}
  & x-1=0 \\
 & \Rightarrow x=0+1 \\
 & \Rightarrow x=1 \\
\end{align}$
Or,
$\begin{align}
  & x+2=0 \\
 & \Rightarrow x=0-2 \\
 & \Rightarrow x=-2 \\
\end{align}$
Hence the values of ‘x’ are $x=0$, $x=1$ and $x=-2$
Wavy Curve Method: After getting the values of x by equating to 0, we have to put them in the number line in proper order and mark +,–,+,… from right to left. Then we have to choose the proper range from the table given below.

Highest degree term	Inequation	solution
positive	>0	‘+’ part
positive	<0	‘– ‘part
negative	>0	‘– ‘part
negative	<0	‘+’ part

Putting the values of ‘x’ in number line

$\begin{align}
  & x\left( x-1 \right)\left( x+2 \right)>0 \\
 & \Rightarrow \left( {{x}^{2}}-x \right)\left( x+2 \right)>0 \\
 & \Rightarrow {{x}^{3}}+2{{x}^{2}}-{{x}^{2}}-2x>0 \\
 & \Rightarrow {{x}^{3}}+{{x}^{2}}-2x>0 \\
\end{align}$
Since the highest degree term of our equation ${{x}^{3}}$ is positive and the inequality is >0
Hence the ‘+’part should be the solution.
So, from the above graph the range is $\left( -2,0 \right)\cup \left( 1,\infty \right)$
This is the solution of the above equation.

Note: In wavy curve method it is necessary to put the values of ‘x’ according to the number line. Otherwise the range may get affected. The alternative ‘+’ and ‘–‘ sings should be given from right to left only, it can’t be from left to right. The range of the solution should be chosen according to the given table.