Solve the inequality x^2(9+x)(x-9)/(x+7)(x-5) is greater than or equal to 0?
I will assume that the denominator is (x+7)(x-5)
or
x^2(9+x)(x-9)/((x+7)(x-5)) ≥ 0
(that is not what you typed)
the critical values are -9, -7, 0, 5, and 9
so pick a value for each domain and evaluate.
We do not need the actual value, just the sign.
1. x< -9 ,
x = -10 ---> +(-)(-)/((-)(-)) = + , so x ≤ 9
2. between -9 and -7
x = -8 ---> +(+)(-)/((-)(-) = - , no good
3. between -7 and 0
x = -5 ---> +(+)(-)/((+)(-) = + , OK
4. the zero ---------- OK
5. between 0 and 5
x = 4 ----> +(+)(-)/((+)(-)) = + , OK
6. between 5 and 9
x = 6 ----> +(+)(-)/((+)(+)) = - , no good
7 x ≥9
x = 10 ----> +(+)(+)/((+)(+)) = + , ok
so x ≤ 9 OR -7 < x < 5 OR x ≥ 9
check out Wolfram
http://www.wolframalpha.com/input/?i=solve+x%5E2%289%2Bx%29%28x-9%29%2F%28%28x%2B7%29%28x-5%29%29+≥+0+
same answer!!
To solve the inequality, we need to find the values of x for which the expression is greater than or equal to zero.
Step 1: Determine the critical points
Critical points occur where the expression is equal to zero or undefined. In this case, we have:
x^2 = 0 (x = 0)
9 + x = 0 (x = -9)
x - 9 = 0 (x = 9)
x + 7 = 0 (x = -7)
x - 5 = 0 (x = 5)
Step 2: Create a sign chart
A sign chart helps us determine the sign of the expression in different intervals. We consider the intervals created by the critical points.
Interval 1: (-∞, -9)
Interval 2: (-9, -7)
Interval 3: (-7, 0)
Interval 4: (0, 5)
Interval 5: (5, 9)
Interval 6: (9, ∞)
Step 3: Test points in each interval
Choose a test point from each interval (excluding the critical points) and substitute it into the expression to determine the sign.
Interval 1: Choose x = -10 (a value less than -9)
(-10)^2(9+(-10))(x-9)/(x+7)(x-5) = (-10)^2(-1)(-19)/(-10+7)(-10-5) = 190/102 = 1.862 > 0 (positive)
Interval 2: Choose x = -8 (a value between -9 and -7)
(-8)^2(9+(-8))(x-9)/(x+7)(x-5) = (-8)^2(1)(-17)/(-8+7)(-8-5) = -272/15 = -18.133 < 0 (negative)
Interval 3: Choose x = -6 (a value between -7 and 0)
(-6)^2(9+(-6))(x-9)/(x+7)(x-5) = (-6)^2(3)(-15)/(-6+7)(-6-5) = 540/33 = 16.364 > 0 (positive)
Interval 4: Choose x = 1 (a value between 0 and 5)
(1)^2(9+1)(x-9)/(x+7)(x-5) = (1)^2(10)(-8)/(1+7)(-4) = -80/32 = -2.500 < 0 (negative)
Interval 5: Choose x = 8 (a value between 5 and 9)
(8)^2(9+8)(x-9)/(x+7)(x-5) = (8)^2(17)(-1)/(8+7)(8-5) = -272/75 = -3.627 < 0 (negative)
Interval 6: Choose x = 10 (a value greater than 9)
(10)^2(9+10)(x-9)/(x+7)(x-5) = (10)^2(19)(1)/(10+7)(10-5) = 190/75 = 2.533 > 0 (positive)
Step 4: Analyze the sign chart
Based on the test points, we can determine the sign of the expression in each interval:
Interval 1: Positive
Interval 2: Negative
Interval 3: Positive
Interval 4: Negative
Interval 5: Negative
Interval 6: Positive
Step 5: Determine the solution
The expression is greater than or equal to zero in the intervals where the sign is either positive or zero. Therefore, the solution is:
x ≤ -9 or -7 < x < 0 or x ≥ 9