Question

solve for x.
4(x - 3)(x - 6) < 0
write the solution using interval notation. use the union symbol ∪ to express the solution as a union of disjoint intervals. finite endpoints of all intervals should be integers. if there are no solutions, use the symbol ∅ for the empty set. use the set notation \\{a\\} to represent an isolated solution a.

Explanation:

Step1: Find critical points

Set each factor equal to 0:
$x-3=0 \implies x=3$
$x-6=0 \implies x=6$
The constant 4 is positive, so it does not affect the inequality sign.

Step2: Test intervals

We test 3 intervals: $(-\infty, 3)$, $(3,6)$, $(6, \infty)$

• For $x=2$ (in $(-\infty,3)$): $4(2-3)(2-6)=4(-1)(-4)=16>0$, does not satisfy $<0$
• For $x=4$ (in $(3,6)$): $4(4-3)(4-6)=4(1)(-2)=-8<0$, satisfies the inequality
• For $x=7$ (in $(6,\infty)$): $4(7-3)(7-6)=4(4)(1)=16>0$, does not satisfy $<0$

Step3: Check endpoints

The inequality is strict ($<0$), so endpoints 3 and 6 are not included.

Answer:

$(3, 6)$