Question 468350: Can someone please help me solve this equation. Thank you.
Solve.
(x+20)(x-15)(x+8)>0
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! Hmmm. This is a "let's think about this" type of problem. Let's try to reason it out by thinking about graphs.
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Suppose we begin by thinking about graphing the left side of the inequality that you were given. In other words, let's pick values of x, substitute them into the left side of the inequality, and calculate the value of the expression for that value of x. We can think of it more as if we had the equation:
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(x+20)(x-15)(x+8) = y
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and were graphing it. What we are looking for is the values of x that will make the left side greater than zero. That means that the corresponding value for y for certain values of x has to be greater than zero. In other words, y has to be positive meaning greater than zero. Think about that logic a bit and see if it doesn't make sense. If we could draw the graph and try to find values of x that have corresponding values of y that are positive (greater than zero and and above the x-axis) we would have the problem solved.
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Next, we note that there are three critical values for x. They are x = -20, x = +15, and x = - 8. Why are these so interesting to us? Because each of these three values make one of the factors on the left equal to zero. That makes the corresponding value of y equal zero and that means the graph touches or crosses the x-axis at that point.
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With that in mind, we need to think about some checking. Just for grins, let's select a value for x that is less than -20. Let's choose x = -100. If x has this value, what is the value of (x + 20)? It's obviously negative -80. So the first factor on the left side is negative. What is the value of (x - 15)? It's -115 so it's negative also. Finally, what's the value of (x + 8)? It's -92 and negative also. So when you multiply these three factors together when x = -100 we get a negative times a negative times a negative, and that results in a negative number. So the product on the left side is negative or less than zero and would graph as point below the x-axis. But we are looking for the product on the left side to be greater than zero (above the x-axis), aren't we? We can now say that for values of x that are less than or equal to -20 we have not satisfied the inequality that we were given in the problem.
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The first time our graph could possibly start into positive values (above the x-axis and meaning that the left side is greater than zero) could occur at x = -20. Why? Because, when x = -20 that's where the graph reaches the x-axis. Remember that at x = -20 the factor (x + 20) = zero so the y value of the graph just equals zero, and for the left side of equation to go from negative territory into the positive region above the x-axis, it must cross the x-axis. So let's try a value of x slightly more than -20. Let's say that x = -19. That being the case, the three factors become (x + 20) = +1, (x - 15) = -34, and (x + 8) = -11. If we multiply these three together we get a positive times a negative times a negative and this equals a positive. That means that the product is now positive and therefore greater than zero. So we have found a region of the graph in which a value of x will make the left side greater than zero.
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The left side will stay greater than zero until the graph again crosses the x- axis and goes down into negative territory. The next time that y equals zero occurs when the value of x = -8 so that the factor (x + 8) = zero. We now know that in the region -20 < x < -8 the graph will be in positive territory and, therefore, the left side (the product of the three factors) is greater than zero.
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But what happens if x is greater than -8. Let's find out. Suppose we set x = 0 which is conveniently greater than x = -8 and is easy to work with. If that happens, the three factors reduce to (+20), (-15), and (+8). This means that we are multiplying a positive by a negative by a positive. The result is negative, and in this region of the graph the value of y is negative, not the positive that we are looking for. We know that when x is greater than -8 the left side is not greater than zero ... the graph is in negative territory below the x-axis.
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The next possible point of change of the graph back into positive territory (above the x-axis) can occur at x = +15. At this point the factor (x - 15) becomes zero and the graph is on the x-axis. Let's see if the graph is in positive territory when x = +16. The three factors become (16 + 20), (16 - 15), and (16 + 8) ... all positive. Therefore, multiplying them makes the result positive which is greater than zero. This means that values of x in the region where x > +15 will cause the left side to be greater than zero.
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There are no other values of x that will make the left side equal zero, meaning making it come back to the x-axis. Therefore the analysis is complete.
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We found that the range x <= 20 is a region that WILL NOT make the left side greater than zero.
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We found that the range -20 < x < -8 is a region that WILL make the left side greater than zero.
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We found that the range -8 <= x <= + 15 is a region that WILL NOT make the left side greater than zero.
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And finally, we found that the range x > +15 is a region that WILL make the left side greater than zero.
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The two answers you are looking for are: -20 < x < -8 and x > +15
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There are other ways to do this problem, but this way is a process that may help you to think about how you can use graphs to understand better what is going on in a mathematical relationship.
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I hope this doesn't confuse you too much. There's a lot of pondering in this method, but if you can see your way through it, it will make you a better mathematician. Good luck.
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