Question

what is the solution set to the inequality 5(x - 2)(x + 4) > 0?
○ {x| x > -4 and x < 2}
○ {x| x < -4 or x > 2}
○ {x| x < -2 or x > 4}
○ {x| x > -2 or x < 4}

Explanation:

Step1: Analyze the coefficient

The inequality is \(5(x - 2)(x + 4)>0\). Since \(5>0\), the sign of the left - hand side is determined by \((x - 2)(x + 4)\). We can use the method of finding the critical points and testing intervals. The critical points are found by setting each factor equal to zero: \(x-2 = 0\) gives \(x = 2\), and \(x + 4=0\) gives \(x=-4\).

Step2: Divide the number line into intervals

The critical points \(x=-4\) and \(x = 2\) divide the real number line into three intervals:

• Interval 1: \(x<-4\). Let's test a value in this interval, say \(x=-5\). Then \((-5 - 2)(-5 + 4)=(-7)\times(-1)=7>0\). So the inequality \((x - 2)(x + 4)>0\) holds for \(x<-4\).
• Interval 2: \(-4
• Interval 3: \(x>2\). Let's test a value in this interval, say \(x = 3\). Then \((3 - 2)(3 + 4)=(1)\times(7)=7>0\). So the inequality \((x - 2)(x + 4)>0\) holds for \(x>2\).

Answer:

\(\{x|x < - 4\text{ or }x>2\}\)