Answer
$(-\infty, -4)\cup(-4,0)\cup(0,\infty)$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given inequality, $
|x^2+4x|\gt0
,$ use the definition of absolute value to analyze the given inequality. Then use concepts of solving quadratic equations to find the values of $x.$
$\bf{\text{Solution Details:}}$
The absolute value of $x$, given by $|x|,$ is the distance of $x$ from $0,$ and hence is always a nonnegative number. Therefore, for any $x,$ the expression at the left, $
|x^2+4x|
,$ is nonnegative. The given inequality is ALWAYS satisfied except when the expression at the left is equal to $0$. Hence, exclude the solutions when $
x^2+4x=0
.$
Factoring the $GCF=x,$ the factored form of the equation above is
\begin{array}{l}\require{cancel}
x(x+4)=0
.\end{array}
Equating each factor to zero (Zero Product Property), the solutions of the equation above are
\begin{array}{l}\require{cancel}
x=0
\\\\\text{OR}\\\\
x+4=0
.\end{array}
Solving the equations results to
\begin{array}{l}\require{cancel}
x=0
\\\\\text{OR}\\\\
x+4=0
\\\\
x=-4
.\end{array}
Hence, the solution set is the set of all real numbers except $x=-4$ and $x=0.$ In interval notation, the solution set is $
(-\infty, -4)\cup(-4,0)\cup(0,\infty)
.$
