Please solve:
(x+3)(x-5) < 0
If the left is less than zero, then either the first parenthesis is negative, or the second, but not both.
I will do one for you, the first paren negative, second positive.
x>5 for second +, x<-3 for the first negative or -3>x>5.
You do the second condition
Do we use the FOIL method here? This is what I got..
(x+3) (x-5) < 0
(x+3<0 and x-5>0) or (x+3>0 and x-5<0)
(x<-3 and x>5) or (x>-3and x<5)
(x<-3 and x>5) or (-3 < x < 5)
-3 < x < 5
Is any of this correct?
To solve the inequality (x+3)(x-5) < 0, we can use the concept of zero-product property and sign analysis. Here's how to do it step by step:
Step 1: Find the critical points
The critical points are the values that make either factor equal to zero. In this case, we have:
x + 3 = 0 => x = -3
x - 5 = 0 => x = 5
Step 2: Create a sign chart
Draw a number line and place the critical points on it:
-∞ | -3 | 5 | +∞
Step 3: Pick test points in each interval
Choose a test point within each interval between the critical points. For convenience, we can choose values smaller than -3, between -3 and 5, and larger than 5, such as:
-4, 0, 6
Step 4: Determine the sign of each factor
Evaluate the sign of each factor using the test points chosen.
For (x + 3):
-4 + 3 = -1 => negative
0 + 3 = 3 => positive
6 + 3 = 9 => positive
For (x - 5):
-4 - 5 = -9 => negative
0 - 5 = -5 => negative
6 - 5 = 1 => positive
Step 5: Determine the sign of the product
Multiply the signs of the factors together to determine the sign of the entire expression in each interval:
For (-∞, -3):
- × - = + => positive
For (-3, 5):
+ × - = - => negative
For (5, +∞):
+ × + = + => positive
Step 6: Find the solution
Since we want the expression (x+3)(x-5) to be less than zero, we look for the intervals where the expression is negative (meaning it is below the x-axis). From the sign chart, we see that this occurs in the interval between -3 and 5 (not including the endpoints). Therefore, the solution to the inequality is:
-3 < x < 5
Note: The inequality is strict (less than) because the expression is not equal to zero at x = -3 and x = 5.