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

Mar 19, 2017

$\left(- \infty , \infty\right)$

#### Explanation:

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

$7 \ge 2 x - 5$
$12 \ge 2 x$
$6 \ge x$

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

$\left(- \infty , 6\right]$

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:

$\frac{3 x - 2}{4} > 4$

$3 x - 2 > 16$
$3 x > 18$
$x > 6$

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:

$\left(6 , \infty\right)$

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 $\left(- \infty , 6\right]$ or the interval $\left(6 , \infty\right)$. 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:

$\left(- \infty , \infty\right)$