Right now we are learning about absolute value inequalities. I have to solve and graph this inequality:

Question

Pretend my paranpheses are absolute value signs

(x-4) + 5 is greater than or equal to 10.

I assume I would take 5 away from each side first. But what would be the correct way to display this on a line, and what would be its set builder notation and interval notation?

Thank you so much! :-)

Answer 1

I took away 5 from each side and added 4 to each side. That got me

(x) is greater than or equal to 14.

Would this make the set builder notation be:

{x such as x is greater than or equal to 4}

And the interval notation be:

(negative infinity, 14] U [14, infinity)?

Thanks! :-)

Answer 2

You have to consider two cases. I will leave out the = sign for convenience.
(1) If x - 4 > 0, then
x - 4 + 5 > 10
x > 9 (which satisfies the x>4 assumption)

(2) If x - 4 < 0, then
If 4 - x + 5 > 10, then
-x > 1
x < -1
(which satisfies the x<4 assumption)