## College Algebra (11th Edition)

$x=\{-4,0\}$
$\bf{\text{Solution Outline:}}$ To solve the given inequality, $|x^2+4x|\le0 ,$ 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 will only be satisfied 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 solutions are $x=\{-4,0\} .$