Video Transcript
Determine the interval where the sign of both of the two functions 𝑓 of 𝑥 is equal to two 𝑥 squared minus seven 𝑥 minus 30 and 𝑔 of 𝑥 equals 𝑥 squared minus three 𝑥 minus 10 is negative for all real numbers.
We can begin by considering the function 𝑓 of 𝑥, which is equal to two 𝑥 squared minus seven 𝑥 minus 30. As we want this to be negative, we want the function to be less than zero. In order to solve the inequality two 𝑥 squared minus seven 𝑥 minus 30 is less than naught, we, firstly, need to find the values where the function is equal to naught. We can do this by factoring or factorizing the left-hand side into two sets of parentheses or brackets. The first terms in our parentheses are two 𝑥 and 𝑥, as two 𝑥 multiplied by 𝑥 is two 𝑥 squared. The second terms must have a product of negative 30. Five multiplied by negative six gives us negative 30.
If we were to expand the set of parentheses, we would get two 𝑥 squared minus 12𝑥 plus five 𝑥 minus 30 which simplifies to two 𝑥 squared minus seven 𝑥 minus 30. We can now solve this by setting each of the brackets equal to zero. Subtracting five from both sides of our first equation gives us two 𝑥 is equal to negative five. Dividing both sides of this equation by two gives us 𝑥 is equal to negative five over two or negative two and a half. Adding six to both sides of our second equation gives us 𝑥 is equal to six. The two solutions to the quadratic equation two 𝑥 squared minus seven 𝑥 minus 30 equals zero are 𝑥 equals negative five over two and 𝑥 is equal to six.
We could demonstrate this on a graph. The quadratic function 𝑓 of 𝑥 intersects the 𝑥-axis at negative five over two and six. We can therefore see that the function is negative — i.e., less than zero — between negative five over two and six. This can be written using inequality signs such that 𝑥 is greater than negative five over two and less than six. Alternatively, we could use interval notation and the open intervals negative five over two to six. Let’s now consider our second function 𝑔 of 𝑥 which is equal to 𝑥 squared minus three 𝑥 minus 10. Once again, we want this to be negative or less than zero. Once again, setting the quadratic equal to zero, we can factor to get 𝑥 minus five multiplied by 𝑥 plus two equals zero. This means that 𝑥 minus five equals zero or 𝑥 plus two equals zero.
Adding five to both sides of the first equation gives us 𝑥 equals five. And subtracting two from both sides of the second equation gives us 𝑥 equals negative two. Once again, this gives us the two values where the function is equal to zero. Sketching the function once again, we see that 𝑔 of 𝑥 is negative between negative two and five. Once again, we could write this using inequality signs or interval notation. The function 𝑔 of 𝑥 is negative when 𝑥 is greater than negative two and less than five.
We want to determine the interval where the sign of both functions is negative. Let’s consider a number line with the four key values negative five over two, negative two, five, and six. We know that 𝑓 of 𝑥 is negative on the open interval negative five over two to six. 𝑔 of 𝑥 is negative on the open interval from negative two to five. As we want both of the functions to be negative, we can see that this occurs between negative two and five. The correct answer is the open interval from negative two to five.
