Question

How do you solve the inequality `abs(4x+7)< -3`?

Answer 1

No solution , since `| 4x + 7| >= 0`

Explanation:

`|4x +7 | < -3` . No solution , since `| 4x + 7| >= 0`

Answer 2

Is this question correct?
Assumption: The question is meant to read `|4x+7|<+3`

With this assumption I get `" "-5/2 < x < -1`

Explanation:

`color(brown)("By definition the outcome of "|4x+7|" can only be positive.")`

`color(brown)("Thus it is not possible for the outcome to be less than negative 3")`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Assumption: The question is meant to read `|4x+7|<+3`

For this to be true we have: `|+-3|=+3`

Case 1: `" "4x+7=-3`

`" "4x=-10`
`" "x=-5/2`

Case 2: `" "4x+7=+3`

`" "4x=-4`
`" "x=-1`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
This very likely defines a range of possible values that are between but not including `x=-1 and x=-5/2" " =>" " -5/2 < x < -1`

Lets test this by selecting values outside this domain

Set `x=-6/2` giving `|4(-6/2)+7|" " =" "|-12+7|=4 " not"<3`

`color(white)()`

Set `x=-9/10` giving
`|4(-9/10)+7|" "=" " |-18/5+7|" "=" "+3 2/5" not" < 3`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(red)("Thus we have: " -5/2 < x < -1`