Please Solve:

(x+3)(x-5)< 0

x^2-2x-15>0

What does this mean? How did you arrive at this conclusion? Please explain thanks!

(x+3)(x-5)< 0 requires that the factors (x+3) and (x-5) have different signs.

Draw the number line, and you will find that from -∞ to -3, both factors are negative. From +5 to +∞, both factors are positive. These intervals will therefore violate the inequality.
The remaining interval ]-3,+5[ (meaning -3<x<5)will satisfy the inequality, as throughout the interval, the two factors have different signs.

To solve the inequality (x+3)(x-5) < 0, you can use the concept of signs or the number line.

Step 1: Determine the critical values
Set each factor equal to zero and solve for x:
x + 3 = 0 --> x = -3
x - 5 = 0 --> x = 5

Step 2: Create a number line and plot the critical values
On a number line, plot the critical values -3 and 5.

-∞ -3 5 +∞

Step 3: Determine the sign of each factor in each interval
For the interval (−∞, −3), pick a test point x = -4 and substitute it into the inequality:
(-4 + 3)(-4 - 5) < 0
(-1)(-9) < 0
9 < 0

Since 9 is not less than 0, the factor (x+3)(x-5) is positive in the interval (−∞, −3).

For the interval (−3, 5), pick a test point x = 0 and substitute it into the inequality:
(0 + 3)(0 - 5) < 0
(3)(-5) < 0
-15 < 0

Since -15 is less than 0, the factor (x+3)(x-5) is negative in the interval (−3, 5).

For the interval (5, +∞), pick a test point x = 6 and substitute it into the inequality:
(6 + 3)(6 - 5) < 0
(9)(1) < 0
9 < 0

Since 9 is not less than 0, the factor (x+3)(x-5) is positive in the interval (5, +∞).

Step 4: Write the solution
The solution to the inequality (x+3)(x-5) < 0 is the interval where the sign of the factor is negative. So, the solution is (-3, 5).