Question

solve the following inequality.
$x^2 - 4x > 0$

select the correct choice below and, if necessary, fill in the answer box.
a. the solution set is \boxed{}.
( type your answer in interval notation. use integers or fractions for any numbers in the expression. )
b. there is no real solution.

Explanation:

Step1: Factor the inequality

Factor \(x^2 - 4x\) to get \(x(x - 4)>0\).

Step2: Find critical points

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

Step3: Test intervals

• For \((-\infty, 0)\), pick \(x=-1\): \((-1)(-1 - 4)=(-1)(-5)=5>0\), so this interval is part of the solution.
• For \((0, 4)\), pick \(x = 1\): \((1)(1 - 4)=(1)(-3)=-3<0\), so this interval is not part of the solution.
• For \((4, \infty)\), pick \(x = 5\): \((5)(5 - 4)=(5)(1)=5>0\), so this interval is part of the solution.

Answer:

The solution set is \((-\infty, 0)\cup(4, \infty)\) (so the correct choice is A with this interval notation).