Question

graph the solution to the following inequality on the number line.
$(x + 5)(x - 4)<0$

Explanation:

Step1: Find the critical points

Set \((x + 5)(x - 4)=0\). Using the zero - product property \(a\times b = 0\) implies \(a = 0\) or \(b = 0\). So \(x+5=0\) gives \(x=-5\), and \(x - 4=0\) gives \(x = 4\).

Step2: Test intervals

We have three intervals to test: \((-\infty,-5)\), \((-5,4)\), and \((4,\infty)\).

• For the interval \((-\infty,-5)\), let \(x=-6\). Then \((-6 + 5)(-6-4)=(-1)\times(-10)=10>0\).
• For the interval \((-5,4)\), let \(x = 0\). Then \((0 + 5)(0-4)=(5)\times(-4)=-20<0\).
• For the interval \((4,\infty)\), let \(x=5\). Then \((5 + 5)(5-4)=(10)\times(1)=10>0\).

Answer:

The solution of the inequality \((x + 5)(x - 4)<0\) is the interval \(-5