First, solve each inequality. I'll solve the first one first.
7
≥
2
x
−
5
12
≥
2
x
6
≥
x
Therefore, x could be any number less than or equal to 6. In interval notation, this looks like:
(
−
∞
,
6
]
The parenthesis means that the lower end is not a solution, but every number above it is. (In this case, the lower end is infinity, so a parenthesis must be used, since infinity is not a real number and so it cannot be a solution.) The bracket means that the upper end is a solution. In this case, it indicates that not only could
x
be any number less than 6, but it could also be 6.
Let's try the second example:
3
x
−
2
4
>
4
3
x
−
2
>
16
3
x
>
18
x
>
6
Therefore, x could be any number greater than 6, but x couldn't be 6, since that would make the two sides of the inequality equal. In interval notation, this looks like:
(
6
,
∞
)
The parentheses mean that neither end of this range is included in the solution set. In this case, it indicates that neither 6 nor infinity are solutions, but every number in between 6 and infinity is a solution (that is, every real number greater than 6 is a solution).
Now, the problem used the word "OR", meaning that either of these equations could be true. That means that either
x
is on the interval
(
−
∞
,
6
]
or the interval
(
6
,
∞
)
. In other words,
x
is either less than or equal to 6, or it is greater than 6. When you combine these two statements, it becomes clear that
x
could be any real number, since no matter what number
x
is, it will fall in one of these intervals. The interval "all real numbers" is written like this:
