Question

solve for x and graph the solution.
$x^2 + 5x + 4 > 0$
plot the endpoints. select an endpoint to change it from closed to open. select the middle of a segment, ray, or line to delete it.
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

Explanation:

Step1: Factor the quadratic inequality

$$x^{2}+5x + 4=(x + 1)(x + 4)>0$$

Step2: Find the critical points

Set \((x + 1)(x + 4)=0\). The solutions are \(x=-4\) and \(x=-1\).

Step3: Test intervals

• For \(x<-4\) (e.g., \(x=-5\)): \((-5 + 1)(-5+4)=(-4)\times(-1) = 4>0\)
• For \(-4
• For \(x>-1\) (e.g., \(x=0\)): \((0 + 1)(0 + 4)=1\times4 = 4>0\)

Answer:

The solution of the inequality \(x^{2}+5x + 4>0\) is \(x<-4\) or \(x>-1\). On the number - line, we have open circles at \(x=-4\) and \(x=-1\), and the graph consists of two rays: one going to the left from \(x = -4\) (not including \(-4\)) and one going to the right from \(x=-1\) (not including \(-1\)).