Question number 7, given that x is between 2 and 3, how many of the following statements are true? We note that we can apply a function to the original statement and obtain each of the following statements if we are a little clever about choosing the function. So we apply to the compound inequality x bigger than 2, less than 3, functions 2x, 3x, x squared, and x squared minus 2x to obtain the above statements. We have just one slight change here to make. If I apply the function 2x, we obtain 2 times 2 for the first term, 2 times x for the second term, and then 2 times 3 for the last one, which is certainly less than 3 times 3. So that statement is still okay. Now each function is increasing on values bigger than 2, and we can even see that this is true for x squared minus 2x, which is a parabola facing up with zeros at the origin and a positive 2. The functions in question preserve the order of inequalities, and so every single expression here in the list obtained from the original statement is also a true statement.

inequalities

functions

compound

parabola

increasing