Solve the inequality x^2(8+x)(x-9)/(x+4)(x-7) is greater than or equal to 0?
x^2(8+x)(x-9)/(x+4)(x-7) >=0
x^2(8+x)(x-9)(x+4)(x-7)>=0
therefore the ranges are x>=9,0<=x<=7,-8<=x<=-4
But x cannot equal -4 or 7
therefore the ranges of x are
x>=9,0<=x<7,-8<=x<-4
Hmm. I get
x in (-oo,-8] U [-4,7] U (9,+oo)
Gotta watch to be sure top and bottom are same sign.
To solve the inequality x^2(8+x)(x-9)/(x+4)(x-7) ≥ 0, we can start by analyzing the signs of the expression at different intervals of the number line.
1. Find the critical points by setting the numerator and denominator equal to zero:
x^2(8+x)(x-9) = 0 and (x+4)(x-7) = 0
The critical points are: x = -4, x = -8, x = 0, x = 7, x = 9.
2. Create sign charts for each factor: (x^2), (8+x), (x-9), (x+4), (x-7).
- For (x^2), the sign does not change because it is always positive.
- For (8+x), the sign changes at x = -8.
- For (x-9), the sign changes at x = 9.
- For (x+4), the sign changes at x = -4.
- For (x-7), the sign changes at x = 7.
3. Combine the sign charts and determine the sign of the overall expression.
At x < -8:
- (x^2) is positive because it is always greater than zero.
- (8+x) is negative because x is less than -8.
- (x-9) is negative because x is less than -8.
- (x+4) is negative because x is less than -8.
- (x-7) is negative because x is less than -8.
So, the overall expression at this interval is positive.
At -8 < x < -4:
- (x^2) is positive because it is always greater than zero.
- (8+x) is negative because x is between -8 and -4.
- (x-9) is negative because x is between -8 and -4.
- (x+4) is positive because x is between -8 and -4.
- (x-7) is negative because x is between -8 and -4.
So, the overall expression at this interval is negative.
At -4 < x < 0:
- (x^2) is positive because it is always greater than zero.
- (8+x) is positive because x is between -4 and 0.
- (x-9) is negative because x is between -4 and 0.
- (x+4) is positive because x is between -4 and 0.
- (x-7) is negative because x is between -4 and 0.
So, the overall expression at this interval is positive.
At 0 < x < 7:
- (x^2) is positive because it is always greater than zero.
- (8+x) is positive because x is between 0 and 7.
- (x-9) is negative because x is between 0 and 7.
- (x+4) is positive because x is between 0 and 7.
- (x-7) is positive because x is between 0 and 7.
So, the overall expression at this interval is negative.
At x > 7:
- (x^2) is positive because it is always greater than zero.
- (8+x) is positive because x is greater than 7.
- (x-9) is positive because x is greater than 7.
- (x+4) is positive because x is greater than 7.
- (x-7) is positive because x is greater than 7.
So, the overall expression at this interval is positive.
4. Based on the sign analysis, we find that the expression is greater than or equal to zero at x < -8, -4 < x < 0, and x > 7.
Therefore, the solution to the inequality x^2(8+x)(x-9)/(x+4)(x-7) ≥ 0 is x ∈ (-∞, -8] ∪ (-4, 0] ∪ (7, ∞).