Question

How do you solve \[{{x}^{2}}>4\] and write the answer as an inequality and interval notation?

Answer 1

Hint: In the given question, we have asked to solve the given inequality for the value of ‘x’. This is a quadratic inequality in one variable as there is only one variable in an equation. Solving quadratic inequality is the same as solving general quadratic equation but there is one more rule i.e. changing the sign of inequality if we multiply both sides by a negative number. To solve this inequality for a given variable ‘ \[x\] ’, we have to undo the mathematical operations such as addition, subtraction, multiplication and division that has been done to the variables. In this way we will get our required answer.

Complete step by step solution:
We have given that,
 \[\Rightarrow {{x}^{2}}>4\]
Taking the square root of both the sides,
 \[\Rightarrow \sqrt{{{x}^{2}}}>\sqrt{4}\]
 \[\Rightarrow \left| x \right|>2\]
It is true either when \[x>2\] or when \[x<-2\]
Therefore,
 \[\Rightarrow -2>x>2\]
In interval notation:
 \[\Rightarrow x\in \left( -\infty ,-2 \right)\cup \left( 2,\infty \right)\]

Note: Students need to remember that while writing down the answer in interval notation form, we will need to think about which interval satisfies this inequality.
In the given question, it is simply all the numbers less than -2, and all the numbers greater than 2. This is the interval from \[-\infty \ to-2\] and \[2\ to\ \infty \] . Note that, here in the given inequality the sign of ‘equals to’ is not given. So the interval does not include 2 or -2 themselves.
Students need to keep in mind that in solving inequality there is one sign changing rule i.e. sign of the inequality will be changed if we multiply both the sides of the inequality by negative number.