can somebody tell me how I am to solve this problem?
x+8)(x-19)(x+3)>0
A. The solution set is {x|___}
B. The solution is all real numbers.
C. There is no solution.
well it is zero if:
x = -8
x = 19
x = -3
If x gets really big positive, expression gets big positive
x big negative, expression big negative
sketch that snake
it is greater than zero between -8 and -3 and also when x is greater than 19
and also
To solve the inequality (x+8)(x-19)(x+3) > 0, we need to consider the signs of each factor.
Step 1: Find the critical points by setting each factor equal to zero and solving for x.
x + 8 = 0 => x = -8
x - 19 = 0 => x = 19
x + 3 = 0 => x = -3
Step 2: Draw a number line and mark the critical points on it (-8, -3, 19).
--(-8)---(-3)---(19)--
Step 3: Choose a test point in each interval created by the critical points and substitute it into the original inequality to determine the sign of the expression.
For the interval (-∞, -8), choose x = -10:
(-10 + 8)(-10 - 19)(-10 + 3) = (-2)(-29)(-7) = 406
Since the result is positive, the expression is greater than zero in this interval.
For the interval (-8, -3), choose x = -5:
(-5 + 8)(-5 - 19)(-5 + 3) = (3)(-24)(-2) = 144
The result is negative, which means the expression is less than zero in this interval.
For the interval (-3, 19), choose x = 0:
(0 + 8)(0 - 19)(0 + 3) = (8)(-19)(3) = -456
Again, the result is negative, indicating that the expression is less than zero in this interval.
For the interval (19, ∞), choose x = 20:
(20 + 8)(20 - 19)(20 + 3) = (28)(1)(23) = 644
The result is positive, meaning the expression is greater than zero in this interval.
Step 4: Determine the solution based on the signs of the expression.
Looking at the number line, we can see that (x+8)(x-19)(x+3) is greater than zero for x in the intervals (-∞, -8) and (19, ∞). Therefore, we can write the solution set as:
A. The solution set is {x|x < -8 or x > 19}
So, the correct answer is A. The solution set is {x|x < -8 or x > 19}.