Question

solve for x.
(x - 4)(x + 6) < 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: \(x - 4 = 0\) gives \(x = 4\); \(x + 6 = 0\) gives \(x = -6\). These are the critical points that divide the number line into intervals: \((-\infty, -6)\), \((-6, 4)\), and \((4, \infty)\).

Step2: Test intervals

• For \((-\infty, -6)\), let's pick \(x = -7\). Then \((-7 - 4)(-7 + 6)=(-11)(-1)=11>0\), so the inequality is not satisfied here.
• For \((-6, 4)\), let's pick \(x = 0\). Then \((0 - 4)(0 + 6)=(-4)(6)= -24<0\), so the inequality is satisfied here.
• For \((4, \infty)\), let's pick \(x = 5\). Then \((5 - 4)(5 + 6)=(1)(11)=11>0\), so the inequality is not satisfied here.

Answer:

\(-6 < x < 4\)