Solve each polynomial inequality in Exercises 1–42 and graph the ...

Question

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. x^2−6x+9<0

Accepted Answer

Hello today we're going to be solving the given polynomial inequality. So what we're given is X squared minus 10. X plus 25 is less than zero. Now. In order to start this process, we want to completely simplify the given polynomial, X squared minus 10 X plus 25 is a fact herbal trin Amiel as this is going to factor into the quantities X minus five times X minus five. And since there are two instances of X -5, we can combine it into a single quantity which is going to be X - squared. Now that this is completely simplified. We want to go ahead and identify the zeros for X. And in order to do that, we're going to take all the factors of X. Set it equal to zero and sulfur X. So this is going to give us X -5 squared is equal to zero. In order to solve for X. We're going to take the square root of both sides of the inequality and that's going to give us X minus five is equal to the square root of zero, which is just zero. And in order to solve for X, we're going to add five to both sides of the equation. And that's going to give us X is equal to five since we've identified the zero for X. We're going to go ahead and plot it on the number line. This number line is going to start a negative infinity, Go to five and then end with positive infinity. And keep in mind that the inequality is using the less than sign since it's using the last end side, we're not going to include the zero in our solution set. And we're going to represent that with an open circle on the number line. Now we want to go ahead and test each region and figure out which region satisfies the inequality. So we're going to take a testing point from the two regions on the number line, plug it into our inequality and whichever one makes inequality true is going to be included in our solution set. So let's go ahead and pick a testing point to the left hand side of five. And that testing point can be when X is equal to zero. When x is equal to zero, we get zero squared minus 10 times zero plus is less than zero. Anything times zero is just going to give us zero. So what we are left with is zero minus zero plus 25 which is going to give us 25 less than zero, which doesn't satisfy the inequality. That means that we're not going to include the region to the left of five. Now, let's pick a testing point to the right of five and this can be when X is equal to six. When x is equal to six, we get six squared minus 10 times six plus 25 is less than zero, Six squared is going to give us 36 and negative 10 times six is going to give us negative 60 Plus 25 is less than zero, and 36 -60 plus 25 is going to give us positive one is less than zero, which is a false inequality. So since this didn't satisfy the inequality, we're not going to include any values to the region of the right of five. Now to interpret what we come up with, we're not including any values that's to the left of five, going to negative infinity. We're not including any values to the right of five that goes to positive infinity. And we're not including the value five itself in our solution set. That means that there are no numbers on the number line that can satisfy the given inequality, meaning that there is no solution to the given inequality. And with that being said, the answer to this problem is going to be d. So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. x^2−6x+9<0

Key Concepts

Polynomial Inequalities

Graphing Solution Sets

Interval Notation

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