Question

Please help<3

Solve and graph the absolute value inequality: |4x + 1| ≤ 5.
A number line with closed dots on −1.5 and 1 with shading going in the opposite directions. B number line with open dots on −1.5 and 1 with shading in between.
C number line with closed dots on −1 and 1 with shading in between.
D number line with closed dots on −1.5 and 1 with shading in between.

Answer 1

if you have < do you use open or closed dots?
if you have \( \le\) do you use open or closed dots?

Answer 2

Thank you for your time.
< closed?
≤ open?
I think the answer is D?
@phi

Answer 3

yes. that helps rule out one of the choices.

when you see absolute value | |, you should think "distance from a middle point"
in other words, it will be an interval. That rules out A

it is down to C or D. to choose, we have to figure out the end points.
You can guess or figure it out.

Answer 4

to figure it out we break the problem into 2 parts:
if what is inside the | | is positive, we can drop the | |
so let's assume the inside is positive:
\( |4x + 1| ≤ 5 \) becomes
\( 4x + 1 ≤ 5 \) and if we solve we get \( x \le 1\)

if the stuff inside is negative, the absolute value makes it positive.
We can have the same effect by multiplying by -1 and dropping the | |
\( -(4x + 1) ≤ 5 \)
to solve we have to be careful. if we multiply or divide by a negative, it changes the order of the <. example -2 < -1 but -1*-2 < -1*-1 gives 2 < 1, which is wrong.
one way around this is always add or subtract.
example: add 4x+1 to both both sides
\[ -(4x+1)+(4x+1) ≤ 5 + (4x+1) \\ 0 \le 4x+6 \\ -6 \le 4x \\ -1.5 \le x \]
that gives the lower point.

Answer 5

-6<=4x<=4
-1.5<=x<=1