Question

solve the following inequality and graph the solution:

$(x + 5)(x + 10) < 0$

write the solution as a compound inequality

and draw the solution

Explanation:

Step1: Find the roots

Set \((x + 5)(x + 10)=0\). Using the zero - product property \(a\times b = 0\) implies \(a = 0\) or \(b = 0\).
If \(x+5=0\), then \(x=-5\). If \(x + 10=0\), then \(x=-10\).

Step2: Test intervals

The roots \(x=-10\) and \(x=-5\) divide the number line into three intervals: \((-\infty,-10)\), \((-10,-5)\), and \((-5,\infty)\).

• For the interval \((-\infty,-10)\), let \(x=-11\). Then \((-11 + 5)(-11+10)=(-6)\times(-1)=6>0\).
• For the interval \((-10,-5)\), let \(x=-7.5\). Then \((-7.5 + 5)(-7.5+10)=(-2.5)\times(2.5)=-6.25<0\).
• For the interval \((-5,\infty)\), let \(x=0\). Then \((0 + 5)(0+10)=5\times10 = 50>0\).

Answer:

The compound inequality is \(-10