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

Question

Solve each quadratic inequality. Give the solution set in interval notation. (x-4)(x + √2) < 0

Accepted Answer

Hello, everyone. We've been asked to provide the solution interval for the given polynomial inequality. The inequality we are given is factor X minus eight, multiplied by a factor X plus radical seven is less than zero. We have four answer choices, all of which are different intervals. And we'll need to see which ones work. So we're gonna start by finding where the breaks in our graph are, which is when the factors equal zero. So X minus eight needs to equal zero as does X plus radical seven, they both equal zero. So to find the X value, I add eight to both sides. So our first endpoint in X value is X equals eight for X plus radical seven equals zero, subtracting radical seven from both sides. So one of our X values is X equals negative radical seven. So those are our two end points. We are going to establish three intervals. We need to check first the negative end. So from negative infinity to our smaller end point, which is negative radical seven, our next interval would be between our two end points. So between negative radical seven and positive eight and our third interval will be from positive eight through positive infinity. That way it covers all the values on the graph starting in our first interval. The lower one I need a value that is less than negative radical seven. Um Without knowing the exact value of negative radical seven, I know it has to be a little bit more than negative three. So I'm gonna pick a number that's less than negative three. So I'm picking negative five. Putting this in my original inequality, I have open parentheses, negative five minus eight closed parentheses multiplied by open parentheses, negative five plus radical seven closed parentheses. And if our point negative five works, this will be less than zero. If this comes out to a true statement, it is our solution false statement, not the solution simplifying what's in the parentheses, negative five minus eight is negative 13. And I'm still gonna multiply that by open parentheses, negative five plus radical seven closed parentheses and still less than zero. Distributing this thir negative 13 into the parentheses. Negative 13 multiplied by negative five is a positive negative 13 multiplied by radical seven. We're gonna write as minus 13 radical seven and this needs to be less than zero. Now, I don't know the exact value of radical seven off the top of my head, but I would look it up to try to find out if this value is gonna come out to be positive or negative. I really don't know quickly without estimating what radical seven is. So for our calculations, we're gonna use radical seven is approximately 2.65. This way, I can see if 65 minus 13 radical seven is even close to being a negative number. I don't know right now. So 65 if I use 2.65 times 13, I get minus 34.45. And for this to be a solution that has to be less than zero, 65 is much larger than 35. So I know this will come out to a positive number. So this is a false statement. So this first interval is not part of our solution. So now we're gonna test the next interval. So the interval between negative radical seven and positive eight, I'm going to pick X equals zero because that is between the two values. And I'd prefer to plug in a zero any time I can. So in my original inequality, I have open parentheses, zero minus eight, closed parentheses multiplied by zero plus radical seven, closed parentheses. And again, for this to be true, this needs to be less than zero. Simplifying in the parentheses, first zero minus eight is negative eight, zero plus radical seven is radical seven. So now I have negative eight multiplied by radical seven needs to be less than zero. Well, I know a positive times a negative is a negative. So negative eight times radical seven will be a negative number and negative numbers are less than zero. So this is a true statement. So this is part of our solution. So so far we know that the interval from negative radical seven to positive eight is part of the solution. We do need to still test our third interval. So from eight to positive infinity, I'm gonna pick X equals 10 because it's greater than eight. And it's a, a nice even number for me. So when I plug that in, I get open parentheses, 10 minus eight, closed parentheses multiplied by open parentheses. 10 plus radical seven closed parentheses. And again, for this to be true, it has to be less than zero within the parentheses. 10 minus eight is two and two is gonna be multiplied by the parentheses. 10 plus radical seven closed parentheses still has to be less than zero. Distributing this in two multiplied by 10 is two, multiplied by radical seven is plus two radical seven. And this has to be less than zero. Again, without doing any further calculations, I can tell you a positive plus a positive will not equal a negative. So positive 20 plus positive two radical seven will not give us a negative number. So this is false. This is not part of our solution. So this means the only section that is part of the solution is the interval from negative radical seven to positive eight. And that's answer choice B have a nice Day.

Solve each quadratic inequality. Give the solution set in interval notation. (x-4)(x + √2) < 0

Key Concepts

Quadratic Inequalities

Factoring Quadratic Expressions

Interval Notation and Sign Analysis

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