Question

How do you solve the inequality $\left( {x + 6} \right)\left( {x - 6} \right) > 0$?

Answer 1

Hint: In this question we are asked to solve the inequality, and these types of questions can be solved by equating each term to zero and the solutions which satisfy the given condition will be the required solution of the given inequality.

Complete step by step solution:
Inequalities are mathematical expressions involving the symbols >, <,$ \geqslant $,$ \leqslant $ To solve an inequality means to find a range, or ranges, of values that an unknown x can take and still satisfy the inequality.
Now the give inequality is $\left( {x + 6} \right)\left( {x - 6} \right) > 0$,
First equate the 1^st term to zero, we get,
$ \Rightarrow x + 6 = 0$,
Now subtract 6 from both sides of the equation we get,
$ \Rightarrow x + 6 - 6 = 0 - 6$,
Now simplifying we get,
$ \Rightarrow x = - 6$,
First equate the 2^nd term to zero, we get,
$ \Rightarrow x - 6 = 0$,
Now subtract 6 from both sides of the equation we get,
$ \Rightarrow x - 6 + 6 = 0 + 6$,
Now simplifying we get,
$ \Rightarrow x = 6$,
Now summarizing above in the table we get,

	$x < - 6$	$ - 6 < x < 6$	$x = 6$	$x = - 6$	$x > 6$ $x > 6$
$x + 6$	-	+	+	0	+
$x - 6$	-	+	0	-	+
$\left( {x + 6} \right)\left( {x - 6} \right)$	+	+	0	0	+

From the table we can see that the given inequality is satisfied when $x < - 6$ and $x > 6$.
So, the solution is $\left( { - \infty , - 6} \right) \cup \left( {6,\infty } \right)$

The solution of the given inequality $\left( {x + 6} \right)\left( {x - 6} \right) > 0$is $\left( { - \infty , - 6} \right) \cup \left( {6,\infty } \right)$.

Note: There are three ways to represent solutions to inequalities: an interval, a graph, and an inequality. Because there is usually more than one solution to an inequality, when you check your answer you should check the end point and one other value to check the direction of the inequality. When we work with inequalities, we can usually treat them similarly to but not exactly as we treat equalities.