Question

Suppose the inequality were `abs(4-x)+15 >14` instead of `abs(4 -x)+ 15 >21`. How would the solution change? Explain.?

Answer 1

Because the absolute value function always returns a positive value, the solution changes from being some of the real numbers `(x< -2; x> 10)` to being all the real numbers `(x inRR)`

Explanation:

It looks like we're starting with the equation

`abs(4-x)+15>21`

We can subtract 15 from both sides and get:

`abs(4-x)+15color(red)(-15)>21color(red)(-15)`

`abs(4-x)>6`

at which point we can solve for `x` and see that we can have `x< -2; x> 10`

So now let's look at

`abs(4-x)+15>14`

and do the same with subtracting 15:

`abs(4-x)+15color(red)(-15)>14color(red)(-15)`

`abs(4-x)> -1`

Because the absolute value sign will always return a value that is positive, there is no value of `x` we can put into this inequality that will make `abs(4-x)<0`, let alone `-1`. And so the solution here is the set of all real numbers, which can be written `x inRR`