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

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