-8 + 2 |4x +6| > 20

Question

Please help. This is on my review packet and I don't quite understand it.

Answer 1

First, isolate the absolute value:

-8 + 2|4x + 6| > 20

2|4x + 6| > 28             add 8 to both sides

|4x + 6| > 14               divide both sides by 2

 

An absolute equation or inequality must be solved twice: once assuming the argument is positive and once assuming the argument is negative.

 

I. 4x + 6 is positive 

4x + 6 > 14

4x > 8          subtract 6 from both sides

x > 2            divide both sides by 4

 

II. 4x + 6 is negative

-(4x + 6) > 14         the absolute value of negative is its opposite

-4x - 6 > 14            distribute

-4x > 20                 add six to both sides

x < -5                    divide both sides by -4 (note order of inequality changes)

 

x < -5 or 2 < x

Answer 2

Margaret, the thing to remember is that the "value" inside the absolute value can be either positive or negative, because both have the same absolute value.

 

With that being said, we start with the positive, so we'll change the absolute value to parentheses.

 

-8 + 2(4x + 6) > 20

-8 + 8x + 12 > 20

                 8x > 16

                   x > 2

 

Now, since 4x + 6 can also be a negative value, we can either make the whole thing negative, -4x - 6, or in this case, change the sign of the 2 we are multiplying by. Both will give the same result.

 

-8 - 2(4x + 6) > 20

    -8 - 8x - 12 > 20

                 -8x > 40

                    x < -5 (dividing by a negative number turns the sign around)

 

So, anything from -5 to 2 will not make the equation true.

 

 

Answer 3

What you must do is ISOLATE THE ABSOLUTE VALUE.

 

You'll get + 2 |4x +6| > 20 + 8

 

|4x +6| > 14.  Now, because this absolute value is a Greater Than situation, you have a DISJUNCTION; 

 

either 4x +6 > 14  OR 4x +6 < -14. You now solve each of these separately; on the numberline, there will be one 'ray' pointing leftward, and another pointing rightward. (These rays exclude their starting points.)