At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.
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 \(|x|\le 5\)?
I'm not sure how to solve that
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.
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.
that makes sense
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?
yeah that makes perfect sense
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.)
That is *one* of the numbers that is three units from zero. Is there another one?
it would be -3
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...
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*.
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!)
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?
Ahhh... you're getting close! Let me ask you this. Can you visit 1.5? Can you visit -2.5?
Right! So, remember, the instructions are: go \(\le\) 3 (miles) away. Can you revise your answer? What neighbors (numbers) can you visit?
Feel free to use words to give your answer for now, like "all the numbers between blah and blah".
Are you there? Do you want a hint?
there are many ways to solve it
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!
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.