Question

solve the inequality \\(dfrac{(x + 6)(x - 4)}{x - 7} \geq 0\\). enter the solution in interval notation.

Explanation:

Step1: Find critical points

Set numerator and denominator zero:

• Numerator: \((x + 6)(x - 4)=0\) gives \(x=-6\), \(x = 4\).
• Denominator: \(x - 7=0\) gives \(x = 7\) (undefined here).

Critical points: \(-6\), \(4\), \(7\). These divide the number line into intervals: \((-\infty, -6)\), \((-6, 4)\), \((4, 7)\), \((7, \infty)\).

Step2: Test intervals

• Interval \((-\infty, -6)\): Pick \(x=-7\).

\(\frac{(-7 + 6)(-7 - 4)}{-7 - 7}=\frac{(-1)(-11)}{-14}=\frac{11}{-14}<0\). Not valid.

• Interval \((-6, 4)\): Pick \(x = 0\).

\(\frac{(0 + 6)(0 - 4)}{0 - 7}=\frac{(6)(-4)}{-7}=\frac{-24}{-7}>0\). Valid.

• Interval \((4, 7)\): Pick \(x = 5\).

\(\frac{(5 + 6)(5 - 4)}{5 - 7}=\frac{(11)(1)}{-2}=\frac{11}{-2}<0\). Not valid.

• Interval \((7, \infty)\): Pick \(x = 8\).

\(\frac{(8 + 6)(8 - 4)}{8 - 7}=\frac{(14)(4)}{1}=56>0\). Valid.

Step3: Check critical points

• \(x=-6\): Substitute, \(\frac{(-6 + 6)(-6 - 4)}{-6 - 7}=0\). So \(x=-6\) is included.
• \(x = 4\): Substitute, \(\frac{(4 + 6)(4 - 4)}{4 - 7}=0\). So \(x = 4\) is included.
• \(x = 7\): Denominator zero, excluded.

Answer:

\([-6, 4] \cup (7, \infty)\)