Perform the indicated operations and simplify.
(z-1)/(z-8)-(z+1)/(z+8)+(z-120)/(z^2-64)
Solve (x+20)(x-12)(x+4)>0
To solve the problem:
1. For the expression (z-1)/(z-8)-(z+1)/(z+8)+(z-120)/(z^2-64):
- First, find a common denominator by multiplying the denominators: (z - 8)(z + 8) = z^2 - 64.
- Now, multiply each term by the appropriate factors to get the common denominator:
(z - 1)(z + 8) - (z + 1)(z - 8) + (z - 120) = (z^2 - z + 8z - 8) - (z^2 + z - 8z + 8) + (z - 120)
Simplify the expression by combining like terms:
z^2 - z + 8z - 8 - z^2 - z + 8z - 8 + z - 120
This further simplifies to:
16z - 256 - 120
16z - 376
2. For the inequality (x+20)(x-12)(x+4) > 0:
- In order to solve this inequality, we need to determine the intervals where the expression is greater than zero.
- Begin by finding the critical points by setting each factor equal to zero:
x + 20 = 0, x - 12 = 0, and x + 4 = 0
Solving these equations gives us: x = -20, x = 12, and x = -4.
- Now, create a number line and mark the critical points on the line (-20, -4, and 12):
-----|-20---|-4---|12---|------
- Choose a value from each interval and substitute it back into the inequality to determine whether it satisfies the condition.
For example, let's choose x = -21:
(-21 + 20)(-21 - 12)(-21 + 4) > 0
(-1)(-33)(-17) > 0
Positive x Negative x Negative = Positive, so it satisfies the condition.
- Repeat this process for another value in the other intervals, such as x = -10 and x = 15.