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← Solve for x - College Algebra

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Showing Revision 2 created 05/25/2016 by Udacity Robot.

  1. So here, I have redrawn our number line, and for each region, I've picked out a
  2. value to test to see whether it satisfies our inequality. First, let's test x
  3. equals negative 11. Now, you might think that we need to actually plug negative
  4. 11 in and find out all the numbers here, but all we really care about is the
  5. sign of this entire expression. We just want to know whether or not it's
  6. positive. So, what I'm going to do is just think about whether or not each of
  7. the factors here, we have 3 of them, is positive or negative. And then, of
  8. course, I also need to take into account the negative sign that's on the
  9. outside. So, I'll write that one first. We have a negative sign on the outside.
  10. If I plug in negative 11 here, negative 11 plus 10 is a negative number, so we
  11. have negative and negative in the numerator. And then in the denominator, we
  12. have a negative number for this factor and a negative number for this factor.
  13. Four negatives multiplied or divided together, gives us a positive number. So,
  14. this region works. Now, let's do the same thing for negative 5. And going
  15. through the same sorts of steps for the next 3, it turns out that only this
  16. region, the first one you found in the area between negative 2 and 2, yield
  17. positive solutions. Let's check the 3 critical values this way also. Notice that
  18. when we plug in negative 10, we end up with 0 for the numerator. That means,
  19. that this entire fraction can never be greater than 0. So, negative 10 is not
  20. part of our solution set. Negative 2 and positive 2 each make the expression on
  21. the left-hand side contain a division by 0, so these cannot be part of our
  22. solution set either. Great. So, we have these two intervals to take into
  23. account. We have the interval negative infinity to negative 10, united with the
  24. interval negative 2 to 2. Awesome. That's our solution. Great job with some
  25. really tough work on quadratic inequalities and rational inequalities. I know
  26. there are a lot of steps in all these problems, but I helped you develop a
  27. little bit of independence in figuring out how to solve these kinds of
  28. inequalities. They're pretty cool, too. I'm going to do one more quick thing and
  29. show you graphically what this inequality might look like.