# Inequalities

• Oct 4th 2013, 06:08 AM
darkangel06
Inequalities
How to solve this inequality

1) -3x^3 >= 81

and which of the following inequalities has a solution

2) x-3<3-x<x-5 or x-3>3-x>x-5
• Oct 4th 2013, 06:24 AM
Plato
Re: Inequalities
Quote:

Originally Posted by darkangel06
How to solve this inequality

1) $\displaystyle -3x^3 \ge 81$

and which of the following inequalities has a solution

2)$\displaystyle x-3<3-x<x-5 \text{ or } x-3>3-x>x-5$

$\displaystyle -3x^3 \ge 81$ is equivalent to
$\displaystyle \\3x^3 \le -81\\x^3\le -27\\x^3\le -3^3$
• Oct 4th 2013, 06:35 AM
HallsofIvy
Re: Inequalities
Quote:

Originally Posted by darkangel06
How to solve this inequality

1) -3x^3 >= 81

What Plato said! And the fact that $\displaystyle x^3$ is an increasing function.

Quote:

and which of the following inequalities has a solution

2) x-3<3-x<x-5 or x-3>3-x>x-5
x- 3< x- 5 is the same as -3< -5 which is not true.

x- 3> x- 5 is the same as -3> -5 which is true.
• Oct 4th 2013, 06:38 AM
votan
Re: Inequalities
Quote:

Originally Posted by darkangel06
How to solve this inequality

1) -3x^3 >= 81

and which of the following inequalities has a solution

2) x-3<3-x<x-5 or x-3>3-x>x-5

detailed solution below
• Oct 4th 2013, 06:42 AM
darkangel06
Re: Inequalities
for the second one It just says to explain which one equality has a solution.
and in the answers the second inequality has a solution but i really dont know how :/
• Oct 4th 2013, 06:56 AM
votan
Re: Inequalities
Quote:

Originally Posted by darkangel06
for the second one It just says to explain which one equality has a solution.
and in the answers the second inequality has a solution but i really dont know how :/

first: x - 3 < 3 - x < x - 5
x < 3 OR x > 4 (OR condition if the double inequality were two separate inequalties) no AND solution

*******3......4******** (* means solultion)

second: x - 3 > 3 - x > x - 5
x > 3 AND x < 4 (AND condition)

.........3*******4.........
• Oct 4th 2013, 07:04 AM
darkangel06
Re: Inequalities
Ok so how did you get x>3 and x< 4 in the second equation
• Oct 4th 2013, 10:11 AM
votan
Re: Inequalities
Quote:

Originally Posted by darkangel06
Ok so how did you get x>3 and x< 4 in the second equation

x - 3 > 3 - x > x - 5

Split this into two inequalities:
x - 3 > 3 - x ==> x + x > 3 + 3 ==> 2x > 6 ==> x > 6/2 = 3 (group all x's on one side and the numbers on the other side)
3 - x > x - 5 == 3 + 5 > x + x ==> 8 > 2x ==> (8/2) > x or x < 4 by reading the inequality from right to left.