Question

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

Answer 1

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 \(|x|\le 5\)?

Answer 2

I'm not sure how to solve that

Answer 3

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.

Answer 4

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.

Answer 5

that makes sense

Answer 6

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?

Answer 7

yeah that makes perfect sense

Answer 8

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.)

Answer 9

3

Answer 10

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

Answer 11

it would be -3

Answer 12

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...

Answer 13

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*.

Answer 14

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!)

Answer 15

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?

Answer 16

1,2,-2,-1

Answer 17

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

Answer 18

yeah

Answer 19

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

Answer 20

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

Answer 21

Are you there? Do you want a hint?

Answer 22

there are many ways to solve it

Answer 23

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!

Answer 24

ah okay.

Answer 25

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.