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Solving a Quadratic Inequality using a Sign Chart - Problem 1
What we are going to be doing, is finding the solutions of this using a sign chart, and then we are going to write the solutions both using set notation and interval notation. So the first thing I could do would be graph this on a graphing calculator. But I don’t want to do that way. I don’t have a graphing calculator handy. I’m going to show you guys how to do it using a sign chart.
First thing I’m going to do is get this quadratic equation and then equal to 0, or our case greater than or equal to 0. Add 6 to both sides. Then I’m going to factor to find my x intercepts, and I’ll see that my x intercepts are going to be at -1 and -6.
So if I were to graph them, those would be the places where my graph would cross the x axis. Let me show you how well you could do this next, using a sign chart on a number line. I’m going to stick -1 and -6 on there. Graph not to scale, let me fix that.
You do want to have them mostly to scale. It doesn’t have to be a perfect, but that was pretty ugly. Even though I’m a brilliant math teacher truth be told, I do make mistakes sometimes but rarely. So on my number line, I’m going to have closed circles because of this inequality sign. It includes the 'or equal to' piece. So I’ll have closed circles on -6 and -1.
The next thing I need to do is test a value out here, test a value in the middle and test a value out here. Test meaning substitute them in to my original inequality or this guy, and see if I get true statements or not. If I do get true statements then those would be part of my solution region, and I would shade them. Let’s test. Let’s say I were to pick x equals -10, and I substitute it into this factored statement. I would have the quantity -10 plus 1 times -10 plus 6.
Now I want to see if that’s greater than or equal to 0. In this case I get a positive answer. This is what I mean by a sign chart. I’m just going to mark that as a positive sign, a plus sign. Now let’s test in here. I wanted to look for things that fall between -6 and -1. I’m going to pick -2 doesn’t matter. So substituting -2 in there, I’ll have -2 plus 1 and then -2 plus 6. In that case I’m going to get a negative value. So I’m going to write here in my sign chart, negative value.
Last region I need to check all numbers that are greater than or equal to -1. I’m going to pick 0. 0 plus 1 is 0, 0 plus 6 is 6, and that is indeed greater than or equal to 0. It’s a positive value. Since I’m looking for things that are greater than or equals to 0, or positive values, that mean is would shade on my number line here, and shade on my number line there.
Any number I select that’s greater than -1, or less than -6, would give me a solution into this original inequality statement.
The last thing I’m going to do is show you guys how you could write these answers in other ways. Maybe your text book asked you to write it in set notation. Remember set notation has curvy brackets and square brackets. So this first piece I would write as going out to negative infinity, and then stopping at -6. -6 is included so we will get a square bracket.
This is similar. I’m going to have a square bracket for -1, going all the way out to positive infinity with the curvy bracket.
So this is the same thing as that written in set notation. The other way you might write it, is using the letters x and the inequality symbols. Like I could write for this piece; x is less than or equal to -6, and also x is greater than or equal to -1. Those two pairings together, these two pairings together, are both different representations of that answer there.
So ask your teacher what he or she wants, or to read your textbook directions really carefully. Make sure you are writing the notation that your text book wants.