Question

How do you solve and write the following in interval notation: `7 ≥ 2x − 5` OR `(3x − 2) / 4>4`?

Answer 1

`(-oo, oo)`

Explanation:

First, solve each inequality. I'll solve the first one first.

`7 >= 2x-5`
`12 >= 2x`
`6 >= x`

Therefore, x could be any number less than or equal to 6. In interval notation, this looks like:

`(-oo, 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:

`(3x-2)/4 > 4`

`3x-2 > 16`
`3x > 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, oo)`

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 `(-oo, 6]` or the interval `(6, oo)`. 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:

`(-oo, oo)`

Final Answer