Practice Exercises Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. 1. (x - 4)(x + 2) > 0 2. (x + 3)(x - 5) > 0 3. (x - 7)(x + 3) ? 0 4. (x + 1)(x - 7) ? 0 5. x² - 5x + 4 > 0 6. x² - 4x + 3 < 0 7. x² + 5x + 4 > 0 8. x² + x - 6 > 0 9. x² - 6x + 9 < 0 10. x² - 2x + 1 > 0 11. 3x² + 10x - 8 ? 0 12. 9x² + 3x - 2 ? 0 13. 2x² + x < 15 14. 6x² + x > 1 15. 4x² + 7x < -3 16. 3x² + 16x < -5 17. 5x ? 2 - 3x² 18. 4x² + 1 ? 4x 19. x² - 4x ? 0 20. x² + 2x < 0 21. 2x² + 3x > 0 22. 3x² - 5x ? 0 23. -x² + x ? 0 24. -x² + 2x ? 0 25. x² ? 4x - 2 26. x² ? 2x + 2 27. 9x² - 6x + 1 < 0 28. 4x² - 4x + 1 ? 0 29. (x - 1)(x - 2)(x - 3) ? 0 30. (x + 1)(x + 2)(x + 3) ? 0 31. x(3 - x)(x - 5) ? 0 32. x(4 - x)(x - 6) ? 0 33. (2 - x)²(x - 7/2) < 0 34. (5 - x)²(x - 13/2) < 0 35. x³ + 2x² - x - 2 ? 0 36. x³ + 2x² - 4x - 8 ? 0 37. x³ - 3x² - 9x + 27 < 0 38. x³ + 7x² - x - 7 < 0 39. x³ + x² + 4x + 4 > 0 40. x³ - x² + 9x - 9 > 0 41. x³ ? 9x² 42. x³ ? 4x² Solve each rational inequality in Exercises 43-60 and graph the solution set on a real number line. Express each solution set in interval notation. 43. (x - 4)/(x + 3) > 0 44. (x + 5)/(x - 2) > 0 57. (x + 4)/(2x - 1) ? 3 59. (x - 2)/(x + 2) ? 2 Practice Plus In Exercises 61–64 61. f(x) = ?... 63. f(x) = ?... Solve each inequality and graph the solution set on a real number line. 65. |x² + 2x...| 67. 3/(x + 3)... 69. (x² - ...)/(x²

Practice Exercises Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. 1. (x - 4)(x + 2) > 0 2. (x + 3)(x - 5) > 0 3. (x - 7)(x + 3) ? 0 4. (x + 1)(x - 7) ? 0 5. x² - 5x + 4 > 0 6. x² - 4x + 3 < 0 7. x² + 5x + 4 > 0 8. x² + x - 6 > 0 9. x² - 6x + 9 < 0 10. x² - 2x + 1 > 0 11. 3x² + 10x - 8 ? 0 12. 9x² + 3x - 2 ? 0 13. 2x² + x < 15 14. 6x² + x > 1 15. 4x² + 7x < -3 16. 3x² + 16x < -5 17. 5x ? 2 - 3x² 18. 4x² + 1 ? 4x 19. x² - 4x ? 0 20. x² + 2x < 0 21. 2x² + 3x > 0 22. 3x² - 5x ? 0 23. -x² + x ? 0 24. -x² + 2x ? 0 25. x² ? 4x - 2 26. x² ? 2x + 2 27. 9x² - 6x + 1 < 0 28. 4x² - 4x + 1 ? 0 29. (x - 1)(x - 2)(x - 3) ? 0 30. (x + 1)(x + 2)(x + 3) ? 0 31. x(3 - x)(x - 5) ? 0 32. x(4 - x)(x - 6) ? 0 33. (2 - x)²(x - 7/2) < 0 34. (5 - x)²(x - 13/2) < 0 35. x³ + 2x² - x - 2 ? 0 36. x³ + 2x² - 4x - 8 ? 0 37. x³ - 3x² - 9x + 27 < 0 38. x³ + 7x² - x - 7 < 0 39. x³ + x² + 4x + 4 > 0 40. x³ - x² + 9x - 9 > 0 41. x³ ? 9x² 42. x³ ? 4x² Solve each rational inequality in Exercises 43-60 and graph the solution set on a real number line. Express each solution set in interval notation. 43. (x - 4)/(x + 3) > 0 44. (x + 5)/(x - 2) > 0 57. (x + 4)/(2x - 1) ? 3 59. (x - 2)/(x + 2) ? 2 Practice Plus In Exercises 61–64 61. f(x) = ?... 63. f(x) = ?... Solve each inequality and graph the solution set on a real number line. 65. |x² + 2x...| 67. 3/(x + 3)... 69. (x² - ...)/(x² - ...) In Exercises 71–... solve each...

Transcript

00:01 In this problem, we are meant to solve all of these inequalities.

00:04 However, that would make the video very long, and it would be too big to upload.

00:09 So i've chosen a few examples out of this block for us to work with.

00:14 So let's get started with number one, where we have x times x minus 4 times x plus 2 is greater than 0.

00:21 So this has already been factored for us.

00:23 So what we can do is split up those factors into two separate inequalities.

00:30 And that's going to be a recurring theme throughout this video.

00:36 So now we just have to solve for x, and so we have to add four to both sides.

00:42 And so we will have x is greater than four.

00:45 And over here we have to subtract two.

00:49 So we have x is greater than minus two.

00:52 Now if you're going to draw this on a number line, draw our number line here, put zero kind of sort of in the middle.

00:59 We have a positive four and a minus two.

01:01 We have to go at least out that far.

01:03 So we have one, two, three, four, there's four, there's two.

01:08 And then we have minus one, minus two.

01:13 So we have, we want all values where x is greater than four.

01:17 So we don't have an equal sign.

01:19 So the actual circle is going to be a bigger circle.

01:22 It's going to be open.

01:23 And then we want all values greater than that.

01:26 So all values this way on the number line.

01:31 Now for the other one, we have x is greater than minus two.

01:35 So here's minus 2, and it's not equal.

01:38 So we want an open loop.

01:39 We want all values that are greater than that one.

01:44 So we go that way.

01:45 So the actual answer to this problem are all values of x that are greater than 4, and remember, they have to also solve the second part of the inequality greater than minus 2.

01:57 And if you notice this inequality sign, the open end is facing towards the x.

02:03 So we want to keep it that way as well.

02:04 So we only need values that are greater than minus 2 and greater than 4 to solve the problem.

02:10 So let's check this out and let's see y for a second.

02:13 So let's put in the value of 0.

02:16 Let's let x equal 0, just to give us a little check.

02:20 So we have 0 minus 4 times 0 plus 2, and this has to be greater than 0.

02:29 So we'll end up with minus 4 times 2, which equals minus 8.

02:35 And that is definitely not greater than zero.

02:39 So this is why they have to be and.

02:42 So the correct visual on the number line, we'll put the zero here.

02:47 And we'll have one, two, three, four.

02:50 We'll put a two there and a four there.

02:52 And keep our open zero.

02:54 And we'll actually have all values going this way to solve the problem.

03:01 Number five, x squared minus five x plus four is greater than zero.

03:06 So this one, we actually have to factor ourselves.

03:10 So let's go ahead and do that.

03:11 We'll have x and x.

03:13 And for our factors of four, we have two times two and one times four.

03:18 And we have to get to a total of minus five.

03:22 Well, if we add these two numbers together, that will actually get us five.

03:25 So we'll put a four over here and a one over here.

03:28 It doesn't actually matter, but we can do it this way.

03:32 So we need a minus five and a minus four.

03:35 So let's put minus signs in both of those because minus four plus minus one equals minus five, and minus one times minus four is equal to positive four.

03:47 So all of these are greater than zero.

03:49 So we'll have x minus one is greater than zero, and x minus four is greater than zero.

03:57 So we want all values of x that are greater than one and all values of x that are greater than four.

04:03 And as we saw on the one above, they have to solve both of these.

04:07 So when we draw it on the number line, you can put our zero here and we'll have one, two, three, four.

04:17 But remember, they have to solve both.

04:19 So we'll have a big circle here to represent values greater than four.

04:24 And you could put a circle here to represent values greater than one.

04:29 But remember, they have to solve both inequalities, and two and three will not solve this inequality.

04:36 So the actual number line should look like this.

04:39 We'll have a zero, and we'll have one, two, three, four, and we'll just have this big circle at the four, and an arrow pointing that way.

04:52 Number 11, three x squared plus 10x minus eight is less than or equal to zero.

04:59 So just like before, let's factor this one out.

05:02 So we'll put three x here and x here.

05:06 Now our factors of eight are eight times one and two times four.

05:11 We have to get to a 10.

05:13 So if we do four times three, that'll get us to 12, and then we can subtract out two, and that'll get us 10.

05:21 So we have to put the four here, and then we have to put the two here.

05:26 We need a positive value and a negative eight, so the positive value will go with plus four, and the negative.

05:34 And the sign will go with 2.

05:35 So now we have 3x minus 2 is less than or equal to 0, and x plus 4 is less than or equal to 0.

05:47 So let's solve this left one first.

05:49 We'll have 3x is less than or equal to positive 2, and x is less than or equal to positive 2 thirds.

06:00 Over here, we'll just have x is less than or equal to minus 4.

06:05 So on the number line, we have a negative 4 and a positive 2 thirds, and we'll put the 0 here...

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