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anonymous

  • one year ago

Solve and graph the absolute value inequality: |4x + 1| ≤ 5.

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  1. anonymous
    • one year ago
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    I can help you with this. First of all, the "4x + 1" inside is making it look harder than it really is. So, let's start a bit easier. Are you able to solve \(|x|\le 5\)?

  2. anonymous
    • one year ago
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    I'm not sure how to solve that

  3. anonymous
    • one year ago
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    Ok, let's start there then. So many people get really confused by absolute value inequalities because they try to memorize lots of different formulas (yuck). Instead, remember that absolute value means distance from zero.

  4. anonymous
    • one year ago
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    When you say |3| = 3, here's what's going on: |3| is asking: "How far away from 0 is the number 3?" The answer is of course 3.

  5. anonymous
    • one year ago
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    that makes sense

  6. anonymous
    • one year ago
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    When you say |-3| = 3, here's what's going on: |-3| is asking: How far from 0 is the number -3" The answer is AGAIN 3. Does this make sense so far?

  7. anonymous
    • one year ago
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    yeah that makes perfect sense

  8. anonymous
    • one year ago
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    Awesome. So, now, if I ask you to solve |x| = 3, here's what you want to think: What numbers (we're calling them x) have a DISTANCE FROM ZERO that is equal to 3. Can you tell me the answer? (Remember ... you can walk away from zero in two directions.)

  9. anonymous
    • one year ago
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    3

  10. anonymous
    • one year ago
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    That is *one* of the numbers that is three units from zero. Is there another one?

  11. anonymous
    • one year ago
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    it would be -3

  12. anonymous
    • one year ago
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    Yes! So, the solution to |x| = 3 is: x = 3 or x = -3 Now, let's go on to the type that you're actually interested in...

  13. anonymous
    • one year ago
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    When you see \(|x|\le 3\), you're really being asked: What are all the numbers whose *distance from zero* is *less than or equal to 3*.

  14. anonymous
    • one year ago
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    So, I want you to imagine yourself standing at zero on a number line. (Maybe this is your "home", and you're driving your Mom crazy!)

  15. anonymous
    • one year ago
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    So your Mom says: please go away for while! But, don't go TOO FAR ... I don't want you to go any more than 3 (miles, maybe) away. Now, you can head out your front door and turn right, or turn left. But, you can't go more than 3 away. What numbers can you visit?

  16. anonymous
    • one year ago
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    1,2,-2,-1

  17. anonymous
    • one year ago
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    Ahhh... you're getting close! Let me ask you this. Can you visit 1.5? Can you visit -2.5?

  18. anonymous
    • one year ago
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    yeah

  19. anonymous
    • one year ago
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    Right! So, remember, the instructions are: go \(\le\) 3 (miles) away. Can you revise your answer? What neighbors (numbers) can you visit?

  20. anonymous
    • one year ago
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    Feel free to use words to give your answer for now, like "all the numbers between blah and blah".

  21. anonymous
    • one year ago
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    Are you there? Do you want a hint?

  22. marcelie
    • one year ago
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    there are many ways to solve it

  23. anonymous
    • one year ago
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    I think he/she may be gone. I've got to get going, so I'll let you take over. I was getting them to the fact that the solution of \(|x| \le 3\) is \(-3 \le x \le 3\). Have a great day!

  24. marcelie
    • one year ago
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    ah okay.

  25. anonymous
    • one year ago
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    If you come back, this page might be helpful: http://www.onemathematicalcat.org/algebra_book/online_problems/solve_simple_abs_val_sen.htm Then, there are also pages for more advanced problems.

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