Solve each quadratic inequality. Give the solution set in interva...

Question

Solve each quadratic inequality. Give the solution set in interval notation. See Example 1. - ( x +√2)(x-3) < 0

Accepted Answer

Hello, everyone. We've been asked to provide the solution interval for the given polynomial inequality. Negative open parentheses. X plus radical five, closed parentheses, multiplied by open parentheses. X minus 11, closed parentheses is less than zero. We have answer choices that are four different intervals. So we'll have to see which interval gives us a true answer. Going back to my original inequality, I note that it starts with a negative sign. So I'm going to divide both sides by negative one and that will flip my inequality and give me what I need to work with. So when I do that, this results in the parentheses, X plus radical five multiplied by the parentheses X minus 11, it will flip the inequality. So it now has to be greater than zero. From here. I want to find the breaks in my graph. So I want to define my intervals, find their end points. So I do each factor set equal to zero. So X plus radical five has to equal zero, an X minus 11 has to equal zero. Finding those X values for X plus radical five equals zero. The X value X equals negative radical five. For X minus 11 equals zero. Again, finding out what X equals X equals positive 11. These are our end points. These are the breaks in the graph. This will be how we find our interval. We're gonna set up three intervals. So first, from negative infinity to negative radical five, next between our two end points. So from negative radical five to positive and then the positive end. So from positive 11 to positive infinity, we're going to pick an X value from each interval to test in the original inequality. So for our first interval from negative infinity to negative radical five, I know negative radical five has to be a little bit less than negative two. So I'm gonna pick a value that's a little bit bigger than that. Oh I'm sorry, a little bit less than that. I'm gonna say X is gonna equal negative five. So plugging that in for X in my original and equality, I have a negative sign multiplied by the parentheses, negative five plus radical five closed parentheses and then open another set of parentheses, negative five minus 11, closed parentheses. And this will have to be less than zero. So I'm gonna simplify everything I can. First, I'm gonna distribute the negative sign into my first set of parentheses. So a negative multiplied by negative five is positive five. A negative multiplied by positive radical five is negative radical five closed parentheses in my second parentheses. Five, I'm sorry. Negative five minus 11 is negative 16. So now I have the parentheses five minus radical five multiplied by negative 16 has to be less than zero. OK. So I'm gonna distribute negative 16 into my binomial when I do so 16, negative 16 times a positive five is a negative negative 16 multiplied by a negative radical five is gonna be a positive 16 radical five. And we wanna know is this less than zero? Now, personally, I do not know what 16 radical five is off the top of my head. So I'm gonna use the approximation that radical five is approximately 2.24. This way, I can see if when I multiply it by 16, I get something that will change this from a negative answer to a positive answer. So multiplying that through I end up with negative 80 plus 35.84 has to be less than zero. Now that I can see what 16 multiplied by radical five is I can tell that negative 80 plus 35.84 will be a negative number thus less than zero and a true statement. So part of our solution. So this is a solution. All right now that we know that we need to test the next interval. So our next interval is from negative radical five to positive 11. I'm going to check the X value or X equals zero because that would fall in between there. So plugging it into my original, I have a negative multiplied by open parentheses. Zero plus radical five closed parentheses, open parentheses, zero minus 11 closed parentheses. And for this to be true, this has to come out to be less than zero. So distributing the negative sign into the first set of parentheses, negative times zero is still zero, negative multiplied by radical five is a negative radical five and then zero minus 11 is negative 11. So we have negative radical five multiplied by negative 11 has to be less than zero. A negative times a negative is going to be a positive. So this will not be a negative number. So this is false and it's not part of our solution. So that section will not be part of the solution last bit from positive 11 to positive infinity. I'm gonna pick the value of 20. So X is gonna be 20. I could pick any value greater than 11. I don't know why I picked 20 but I did so negative times in parentheses. 20 plus radical five, closed parentheses, open parentheses 20 minus 11. And this has to come out to be less than zero. So as I've done previously, I'm going to multiply the negative sign into the first set of parentheses. So negative multiplied by 20 is negative 20. A negative multiplied by radical five is negative radical five, putting those back in parentheses and then 20 minus 11 is nine. And this again, needs to be less than zero. So we have negative 20 minus radical five multiplied by nine. So I'm gonna distribute the nine into my binomial nine, multiplied by negative 20 is negative nine, multiplied by negative radical five is gonna be minus nine radical five. And all of this needs to be less than zero. Well, without doing any real calculations, I know a negative minus a negative will give me a negative. So when X is 20 this is a true statement. So again, part of our solution. So this means, and we have two intervals that gave us true statements. The interval from negative infinity to negative radical five and the interval from 11 to positive infinity. So our solution is both of those intervals which is answer choice a the interval from negative infinity to negative radical five union with the interval 11 to positive infinity have a nice day.

Solve each quadratic inequality. Give the solution set in interval notation. See Example 1. - ( x +√2)(x-3) < 0

Key Concepts

Quadratic Inequalities

Interval Notation

Sign Analysis

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