If x < -7, simplify |4 - |3 + x||

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hala718's profile pic

hala718 | High School Teacher | (Level 1) Educator Emeritus

Posted on

x < -7

simplify |4 - |3 + x||

First we will simplify the expression l 3 + x l

Since x < -7

Then we will add 3 to both sides:

==> x + 3 < -7 + 3

==> x+ 3 < -4

Then we conclude that ( x+ 3) is a negative number:

Then l x + 3 l = - (x+3)

Now let us substitute in the expression:

 l 4 - l x+3 l = l 4 - (-(x+3) l

                   = l 4 +x + 3 l

                    = l x +7 l

We know that x < -7

==> x + 7 < -7 + 7

==> x+7 < 0

Then (x+ 7) is a negative number:

==> l x + 7l = - (x+7)

Then :

l 4 - l 3+xl l = -x -7

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

If x<-7, we'll add 3 both sides:

3 + x < 3 - 7

3 + x < -4

We'll consider the absolute values both sides:

|3+x|<|-4|

|3+x|<4

We'll multiply by -1 both sides:

-|3+x| > -4

We'll add 4 both sides:

4 - |3 + x| > 4 - 4

4 - |3 + x| > 0

|4 - |3 + x|| > 0

Now, we'll solve |3+x|<4:

-4<3+x<4

-4<3+x

We'll add 4 both sides:

0 < 7+x

x>-7

We'll solve the right inequality;

3+x<4

x < 4 - 3

x < 1

The values of x are bhounded by the end values:

-7 < x < 1

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