Can this be solved: | 4m+8 | <12 ?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted on

You need to use the following absolute value property, such that:

`|a| < b => -b < a < b => a> -b and a < b => {(-b < a),(a < b):}`

Reasoning by analogy yields:

`| 4m+8 | <12 => -12 < 4m + 8 < 12 => {(4m + 8 > -12),(4m + 8 < 12):}`

`{(m + 2 > -3),(m + 2 < 3):} => {(m> -3 - 2),(m < 3 - 2):}`

`{(m> -5),(m < 1):} => {(m in (-5, +oo)),(m in (-oo,1)):} => m in (-5,oo) nn (-oo,1) => m in (-5,1)`

Hence, evaluating the solution to the absolute value inequality yields `m in (-5,1).`

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crmhaske | College Teacher | (Level 3) Associate Educator

Posted on

You're right.  In my haste I neglected that |4(-6)+8| > 12.  Thank you.

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crmhaske | College Teacher | (Level 3) Associate Educator

Posted on

The line brackets around the expression denote the absolute value; therefore, the solution can be either positive or negative:

`4m+8lt+-12`

Consequently, there are two possible solutions for m:

`4m+8lt12`

`4mlt12-8`

`4mlt4`

`mlt1`

`4m+8lt-12`

`4mlt-12-8`

`4mlt-20`

`mlt-5`

Therefore, m must be less than -5 or less than 1 in order to satisfy the expression.  As `-5lt1` there is one solution:

`mlt1`

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