Help with finding least upper bounds and greatest lower bounds?

[mgnymph]

Homework Statement

Solve the following inequalities and express the solution(s) in interval notation and set builder notation. For each of these, state the least upper bound and greatest lower bounds, if these exist.

Homework Equations

i) x^3 + x^2 > 2x


ii) l 2 - x l =< 4
(modulus of 2 - x is greater than or equal to 4)

The Attempt at a Solution

So for (i), I factorised so

x^3 + x^2 - 2x > 0
x(x^2 + x -2) > 0
x(x+2)(x-1) > 0

I am really unsure what interval notation and set builder notation are, but I think...
Interval notation: x E (-2,0) U (1 , infinity)
Set builder notation: {x : -2 < x < 0 or x > 1}

And I don't know how to find the bounds...

(ii) l 2 - x l =< 4
-4 =< 2 - x =< 4

-6 =< x =< 2

interval notation: x E [-2,6]
set builder notation: {x: -2 =< x =< 6}

and... i don't know how to find the least upper bounds/greatest lower bounds for this either.

=/

 

[Mark44]

Homework Statement

Solve the following inequalities and express the solution(s) in interval notation and set builder notation. For each of these, state the least upper bound and greatest lower bounds, if these exist.

Homework Equations

i) x^3 + x^2 > 2x


ii) l 2 - x l =< 4
(modulus of 2 - x is greater than or equal to 4)

The Attempt at a Solution

So for (i), I factorised so

x^3 + x^2 - 2x > 0
x(x^2 + x -2) > 0
x(x+2)(x-1) > 0

I am really unsure what interval notation and set builder notation are, but I think...
Interval notation: x E (-2,0) U (1 , infinity)
Set builder notation: {x : -2 < x < 0 or x > 1}

This looks fine. For the least upper bound and greatest lower bound, I think the problem is asking what they are for each of the two intervals. For the interval (-2, 0), the glb is -2 and the lub is 0. Since the interval is open, these bounds are not included in the interval.
For the interval (1, infinity), there is no upper bound, so there isn't a least upper bound. The greatest lower bound is 1, which is not an element of this interval.

And I don't know how to find the bounds...

(ii) l 2 - x l =< 4
-4 =< 2 - x =< 4

So far so good in your inequality above. You've made a mistake in the one below, though. Add -2 to each member of the inequality and you get -6 <= -x <= 2. If you then multiply each member of the inequality by -1, what happens to the direction of the inequality symbols?

-6 =< x =< 2

interval notation: x E [-2,6]
set builder notation: {x: -2 =< x =< 6}

and... i don't know how to find the least upper bounds/greatest lower bounds for this either.

=/

I think they'll be the two endpoints of the correct interval.