Find all values of x that satisfy x(x+7.5) > 38.5. Give your answer as an interval. Please enter your response in interval notation. Refer t

x(x+7.5) > 38.5        simplifying, we have

x^2 + 7.5x > 38.5    subtract 38.5 from both sides

x^2 + 7.5x - 38.5 > 0

This is a little difficult to think of the factorization, here ...(but, it actually does factor, believe it or not !!)...the easiest way to solve this is to look at a graph.....https://www.desmos.com/calculator/8efn8ekufz

Note that the intervals that make this true are (-∞, -11) and (3.5, ∞)

I have answers that I know are incorrect. Please give the answers as they are asked. For example, if it asks for interval, write it as interval. If it is interval notation, write it in interval notation. Thank you!

Answer:  (-∞, -10.965)  ∪  (3465,+∞)

Because:

x(x + 7.5) > 38

x² + 7.5x - 38 > 0

First, find the solutions of  x² + 7.5x - 38 = 0

I used the quadratic formula and got  x =  3.465  and  x  =  -10.965

These two numbers divide the x-axis into three groups.

So, there are three groups  (x <  -10.965)        (-10.965 < x < 3.465)       (x > 3.465)

Try any number from the first group; I'll choose -20; and put it back into the original inequality:

       x(x + 7.5) > 38     ===>     -20(-20 + 7.5)  =  250 and that's > 38 go this section works.

Try any number from the second group; I'll choose 0; and put it back into the original inequality:

       x(x + 7.5) > 38     ===>     0(0 + 7.5)  =  0 and that's not > 38 go this section doesn't works.

Try any number from the third group; I'll choose 10; and put it back into the original inequality:

       x(x + 7.5) > 38     ===>     10(10 + 7.5)  =  175 and that's > 38 go this section works.

Since the first group and the third group both work, the answer is:  x < -10.965  or  x > 3.465.

how wouldyou write this in interval notation?

OK...we can use

(-∞ < x < -11) U (3.5 < x < ∞)

Thumbs up to Chris for spotting an error

and thumbs up to Geno for all his effort.    :)