Question

solve for x.
(x - 2)(x - 4) < 0
write a compound inequality like 1 < x < 3 or like x < 1 or x > 3.

Explanation:

Step1: Find critical points

Set each factor equal to zero: \(x - 2 = 0\) gives \(x = 2\); \(x - 4 = 0\) gives \(x = 4\). These are the critical points that divide the number line into intervals: \((-\infty, 2)\), \((2, 4)\), and \((4, \infty)\).

Step2: Test intervals

• For \(x < 2\) (e.g., \(x = 0\)): \((0 - 2)(0 - 4)=(-2)(-4) = 8>0\), so not in solution.
• For \(2 < x < 4\) (e.g., \(x = 3\)): \((3 - 2)(3 - 4)=(1)(-1)= - 1<0\), so in solution.
• For \(x > 4\) (e.g., \(x = 5\)): \((5 - 2)(5 - 4)=(3)(1)=3>0\), so not in solution.

Answer:

\(2 < x < 4\)