# Inequality

• Jan 14th 2010, 05:23 PM
ashleydevos
Inequality
Help!! I need help finding x.
2x – (7 + x) < x

2x - 7 - x <= x
x - 7 <= x

The x cancels out, so is the answer that circle with the diagonal line through it?
• Jan 14th 2010, 05:35 PM
skeeter
Quote:

Originally Posted by ashleydevos
Help!! I need help finding x.
2x – (7 + x) < x

2x - 7 - x <= x
x - 7 <= x

The x cancels out, so is the answer that circle with the diagonal line through it?

$\displaystyle x-7 \le x$

you did not finish, subtract $\displaystyle x$ from both sides ...

$\displaystyle -7 \le 0$

a true statement ... therefore the inequality is true for all $\displaystyle x$
• Jan 14th 2010, 05:36 PM
Bacterius
$\displaystyle 2x - (7 + x) \leq x$

$\displaystyle 2x - 7 - x \leq x$

$\displaystyle x - 7 \leq x$

$\displaystyle -7 \leq 0$

So this equality holds for any $\displaystyle x$, because $\displaystyle -7$ is indeed smaller than $\displaystyle 0$.

EDIT : erf Skeeter beat me to it :D
• Jan 14th 2010, 07:01 PM
ashleydevos
How do you write it in interval notation then? [-7<= ∞)???
• Jan 14th 2010, 07:20 PM
Bacterius
No, no !

Regardless of the value of $\displaystyle x$, the inequality holds, so I guess you could write this $\displaystyle x = \mathbb{R}$, or :

$\displaystyle \forall x \in \mathbb{R}$, $\displaystyle 2x - (7 + x) \leq x$

A less elegant but more direct way to write it could be : $\displaystyle x = (-\infty, +\infty)$, $\displaystyle x \in \mathbb{R}$.