Question

graph the solution to the following inequality on the number line

$x^{2}>-4x$

note that you can use the aleks graphing calculator to help get your answer.

Explanation:

Step1: Rewrite the inequality

Start with \(x^{2}>-4x\). Move all terms to one side to get \(x^{2}+4x > 0\). Factor the left - hand side: \(x(x + 4)>0\).

Step2: Find the critical points

Set \(x(x + 4)=0\). Using the zero - product property \(a\times b = 0\) implies \(a = 0\) or \(b = 0\). So \(x=0\) or \(x+4=0\) (i.e., \(x=-4\)). These two points \(x=-4\) and \(x = 0\) divide the real - number line into three intervals: \((-\infty,-4)\), \((-4,0)\), and \((0,\infty)\).

Step3: Test the intervals

• For the interval \((-\infty,-4)\), let \(x=-5\). Then \(x(x + 4)=(-5)\times(-5 + 4)=(-5)\times(-1)=5>0\).
• For the interval \((-4,0)\), let \(x=-1\). Then \(x(x + 4)=(-1)\times(-1 + 4)=(-1)\times3=-3<0\).
• For the interval \((0,\infty)\), let \(x = 1\). Then \(x(x + 4)=1\times(1 + 4)=5>0\).

Since the inequality is \(x(x + 4)>0\), the solution of the inequality \(x^{2}>-4x\) is the union of the intervals where the expression \(x(x + 4)\) is positive, which is \(x<-4\) or \(x > 0\).

Answer:

The solution is \(x < - 4\) or \(x>0\). On the number - line, there should be an open circle at \(x = - 4\) and \(x = 0\), and the line should be shaded to the left of \(x=-4\) and to the right of \(x = 0\).