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- anonymous

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

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- anonymous

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

- schrodinger

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- anonymous

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 \(|x|\le 5\)?

- anonymous

I'm not sure how to solve that

- anonymous

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.

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- anonymous

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.

- anonymous

that makes sense

- anonymous

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?

- anonymous

yeah that makes perfect sense

- anonymous

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.)

- anonymous

3

- anonymous

That is *one* of the numbers that is three units from zero. Is there another one?

- anonymous

it would be -3

- anonymous

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...

- anonymous

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*.

- anonymous

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!)

- anonymous

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?

- anonymous

1,2,-2,-1

- anonymous

Ahhh... you're getting close! Let me ask you this. Can you visit 1.5? Can you visit -2.5?

- anonymous

yeah

- anonymous

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

- anonymous

Feel free to use words to give your answer for now, like "all the numbers between blah and blah".

- anonymous

Are you there? Do you want a hint?

- marcelie

there are many ways to solve it

- anonymous

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!

- marcelie

ah okay.

- anonymous

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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