Question

solve for x.
3(x - 3)(x - 5) < 0
write a compound inequality like 1 < x < 3 or like x < 1 or x > 3.

Explanation:

Step1: Analyze the coefficient

The coefficient of the quadratic part is \(3\), which is positive. So the sign of the expression \(3(x - 3)(x - 5)\) is determined by the sign of \((x - 3)(x - 5)\).

Step2: Find critical points

Set \((x - 3)(x - 5)=0\), then \(x - 3 = 0\) or \(x - 5 = 0\), so \(x = 3\) or \(x = 5\). These are the critical points that divide the number line into three intervals: \((-\infty, 3)\), \((3, 5)\), and \((5, +\infty)\).

Step3: Test intervals

• For \(x < 3\) (e.g., \(x = 0\)): \((0 - 3)(0 - 5)=(-3)\times(-5)=15>0\), so \(3(x - 3)(x - 5)>0\) in this interval.
• For \(3 < x < 5\) (e.g., \(x = 4\)): \((4 - 3)(4 - 5)=(1)\times(-1)=-1<0\), so \(3(x - 3)(x - 5)<0\) in this interval.
• For \(x > 5\) (e.g., \(x = 6\)): \((6 - 3)(6 - 5)=(3)\times(1)=3>0\), so \(3(x - 3)(x - 5)>0\) in this interval.

Answer:

\(3 < x < 5\)