solve the inequality algebraically.\n\\( \\frac { ( x - 20 ) ( x + 20 ) } { x } \\leq 0 \\)\nmove all terms…
solve the inequality algebraically.\n\\( \\frac { ( x - 20 ) ( x + 20 ) } { x } \\leq 0 \\)\nmove all terms to the left side of the inequality and then define the left side of the inequality as \\( f ( x ) \\). then list the intervals determined by the real zeros of \\( f \\) and the real numbers for which \\( f \\) is\nundefined. complete the following table.\n(type your answers in interval notation. use ascending order.)
Answer
Explanation:
Step1: Analyze the sign of the function
We have the inequality (\frac{(x - 20)(x+20)}{x}\leq0). The critical points are (x=- 20), (x = 0), and (x = 20) (where the numerator or denominator is zero).
Step2: Test intervals
- For (x\in(-\infty,-20)), let (x=-21). Then (\frac{(-21 - 20)(-21 + 20)}{-21}=\frac{(-41)(-1)}{-21}=-\frac{41}{21}<0)
- For (x\in(-20,0)), let (x=-1). Then (\frac{(-1 - 20)(-1 + 20)}{-1}=\frac{(-21)(19)}{-1}=399>0)
- For (x\in(0,20)), let (x = 1). Then (\frac{(1 - 20)(1+20)}{1}=\frac{(-19)(21)}{1}=-399<0)
- For (x\in(20,\infty)), let (x = 21). Then (\frac{(21 - 20)(21 + 20)}{21}=\frac{(1)(41)}{21}=\frac{41}{21}>0)
Step3: Consider the critical points
When (x=-20), (\frac{(-20 - 20)(-20+20)}{-20}=0). When (x = 20), (\frac{(20 - 20)(20+20)}{20}=0). When (x = 0), the function is undefined.
Answer:
((-\infty,-20]\cup(0,20])