Solve
(x+17)(x-20)(x+8)>0
The solution set is (x )?
Use at least one inequality or compound inequality to express your answer. For answers with more than one inequality, separate the inequalities by a comma or the word 'or'. Type R is the answer is all real numbers. Type N if there is no solution.
Goodness. To be greater than zero, all () have to be positive, or two negative.
If x>-17, the first is postive, the second negative until x>20, and the third is negative until x>-8.
So the solution for this parenthesis is x is >-17 and less than -8.
Try the other two, we will be happy to critique yourthinking.
This is what I came up with for a final answer
{x straight up and down line-17<x<-8 or x>20}
To solve the inequality (x+17)(x-20)(x+8)>0, we need to determine the intervals on the number line where the expression is greater than zero.
To do this, we can analyze the signs of each factor: (x+17), (x-20), and (x+8).
Case 1: (x+17)>0, (x-20)>0, and (x+8)>0
In this case, all three factors are positive. To find the interval where this condition holds true, we can solve each inequality independently:
x+17 > 0
x > -17
x-20 > 0
x > 20
x+8 > 0
x > -8
Since all three inequalities need to be simultaneously satisfied, we take the intersection of their intervals, which gives us x > 20.
Case 2: (x+17)<0, (x-20)<0, and (x+8)<0
In this case, all three factors are negative. To find the interval where this condition holds true, we solve the inequalities independently:
x+17 < 0
x < -17
x-20 < 0
x < 20
x+8 < 0
x < -8
Again, taking the intersection of these intervals, we get x < -17.
Therefore, the solution set for the inequality (x+17)(x-20)(x+8)>0 is x < -17 or x > 20.
To solve the inequality (x+17)(x-20)(x+8)>0, we can use the concept of interval notation and test different intervals to find the solution set. Here's how we can approach it:
Step 1: Find the critical points.
The critical points are the values of x that make any of the factors of the expression equal to zero. In this case, we have:
x + 17 = 0 => x = -17
x - 20 = 0 => x = 20
x + 8 = 0 => x = -8
Step 2: Create a number line.
We'll create a number line and mark the critical points (-17, -8, 20) on it. This will help us determine intervals to test.
-∞ |----|----|----|----|----|----|----|----|+∞
-17 -8 20
Step 3: Test the intervals.
Now, we will test each interval on the number line to check which intervals satisfy the given inequality. We can choose a test point from each interval and substitute it into the inequality.
For the interval (-∞, -17):
Let's choose x = -18 as a test point.
Substituting x = -18 into the inequality: (-18+17)(-18-20)(-18+8) > 0
(-1)(-38)(-10) > 0
380 > 0
Since 380 is greater than zero, the interval (-∞, -17) satisfies the inequality.
For the interval (-17, -8):
Let's choose x = -10 as a test point.
Substituting x = -10 into the inequality: (-10+17)(-10-20)(-10+8) > 0
(7)(-30)(-2) > 0
420 > 0
The interval (-17, -8) also satisfies the inequality.
For the interval (-8, 20):
Let's choose x = 0 as a test point.
Substituting x = 0 into the inequality: (0+17)(0-20)(0+8) > 0
(17)(-20)(8) < 0
The test point (-8, 0) does not satisfy the inequality.
For the interval (20, +∞):
Let's choose x = 21 as a test point.
Substituting x = 21 into the inequality: (21+17)(21-20)(21+8) > 0
(38)(1)(29) > 0
The test point (20, +∞) satisfies the inequality.
Step 4: Express the answer using inequalities.
The intervals (-∞, -17) and (-17, -8) satisfy the inequality while the interval (-8, 20) does not. Therefore, the solution set can be expressed as:
x < -17 or -8 < x < 20
In interval notation, the solution set is:
(-∞, -17) ∪ (-8, 20)