Use the technique described in Exercises 87–90 to solve each ineq...

Question

Use the technique described in Exercises 87–90 to solve each inequality. Write the solution set in interval notation. x^2 - 9x + 20 < 0

Accepted Answer

Hello, everyone. In this video, we're going to be looking at this practice problem which states find the solution for the given quadratic inequality express the answer in interval notation. And the quadratic inequality is X to the power of two minus 17 X plus 60 less than zero. And were given four answer choices A B C and D and in each of these answer choices, there's an interval and the numbers seem to focus on five or 12, whether it be negative 12 and negative five or variations of that. And you can see that A and D have the same interval with the exception that D has brackets instead of parentheses. So in order to figure out which of these is correct, let's take a look at our inequality. So we have X to the power of two minus 17 X plus 60 is less than zero. Now, this is very similar to a quadratic equation with the exception that it's just a quadratic inequality instead of the equal to. So when I want to find the solution for this inequality, I'm going to try to solve for X and I'm going to try to solve for X by factoring this quadratic equation. So my first terms of my factors need to multiply to get me X squared. So I'll have X times X gives me X squared. So I can write X as the first term. And my second terms need to multiply to give me positive 60. So when we think of factors of 60 there's 6 10, 5 12 and the negative versions of that. So for this problem, I'm going to try X -5 And X -12 as my factors. And we can check to make sure that when we multiply these two factors together, we get back our original quadratic equation. So I have X multiplied by X is X squared X multiplied by negative 12 is negative 12. X negative five, multiplied by X is negative five X and negative five multiplied by negative 12 is positive 60. So once I combine these middle terms, I'm left with negative 17 X. So I know that my factors are good. So I'm going to use them to solve for X. So my first factor I have X -5 is equal to zero. If I solve for X by adding five, I get that X is equal to five. My second factor is X -12, equal to zero. I solve for X by adding 12 to both sides, I get X is equal to 12. So you may be thinking that these are the final solutions and they would be if we were dealing with a quadratic equation. But because we're dealing with an inequality, these values become the points where we need to break up the X axis and look at the different intervals. So if I were to draw a number line Draw, take care at five and and say that this is the x axis, I care about this, these intervals where I have negative infinity 25, Then I have five 2 12 And then my final interval 12 to positive infinity. So I need to look at points in these intervals and see if my inequality is true at those points. So for my first interval, negative infinity to five, I'm going to test X equals zero. So I plug in zero for X I get zero squared minus 17, multiplied by zero plus 60 is less than zero while zero squared is zero and 17 times zero is zero. So I'm left with 60 is less than zero. And in this case, that is not true. So I can exclude this interval from my solution set. If I look at my next interval, I have five 2 12, I'm going to test the value X equals six. So if I plug in six, I have six to the power of two -17 multiplied by six plus 60 is less than zero 6 to the power of two is 6 times six, which is 36 and 17 times six is one oh to, so I have 36 minus 102 plus 60 is less than zero. And when I combine all these numbers, I get negative six is less than zero. And that is true. So I can include this interval in my solution set. And the final interval we need to test is from 12 to positive infinity. I'm going to plug in X equals 13 sort of 13 to the power of two minus 17, multiplied by 13 plus 60. Less than zero, 13 Squared is -17 times 13, which is equal to 221 Plus is less than zero. And if I combine all of these, I get the eight is Less than zero. Well, in this case, that is not true. So I can exclude this interval from my solution set. So I know that the only interval I have that gave me a true Statement is from 5 to positive 12. So that's here on this Number line. And now I need to know if I need to include these points. If x equals five And x equals 12, make my inequality true. So if I test X equals five, I have five to the power of two minus 17 times five plus 60 is less than 0, 25 squared is 25 and 17 times five is 85-plus 60 less than zero. Still, if I combine all these numbers, I get zero is less than zero. And that is not true. So I can exclude five from my solution set. And next I can test X equals 12. So if I test X equals 12, I have 12 to the power of two minus 17 times 12 plus 60 is less than zero. 12. Squared. Is this 144 - times 12, which is plus 60 less than zero. So if I combine these numbers 144 - plus 60 is zero. So this says zero is less than zero, which is not true. So I can exclude x equals 12 from my solution set. So if I were to do demonstrate that I don't include five or 12 in my solution set on this number line, it would be an open circle. And if I write that in interval notation, I have parentheses, my lower limit which is five Up to my upper limit which is 12 and another parentheses because I can't equal those numbers exactly. So my final solution set is 5-12. And if we look at the answer choices, you can see that a also has 5-12 with parentheses. So hopefully this video helped and see you next time.

Use the technique described in Exercises 87–90 to solve each inequality. Write the solution set in interval notation. x^2 - 9x + 20 < 0

Key Concepts

Quadratic Inequalities

Factoring Quadratics

Interval Notation

Watch next