<h2><em><u>Answer:</u></em></h2><h2><em><u>First, solve each inequality. I'll solve the first one first.
</u></em></h2><h2><em><u>
</u></em></h2><h2><em><u>7
</u></em></h2><h2><em><u>≥
</u></em></h2><h2><em><u>2
</u></em></h2><h2><em><u>x
</u></em></h2><h2><em><u>−
</u></em></h2><h2><em><u>5
</u></em></h2><h2><em><u>
</u></em></h2><h2><em><u>12
</u></em></h2><h2><em><u>≥
</u></em></h2><h2><em><u>2
</u></em></h2><h2><em><u>x
</u></em></h2><h2><em><u>
</u></em></h2><h2><em><u>6
</u></em></h2><h2><em><u>≥
</u></em></h2><h2><em><u>x
</u></em></h2><h2><em><u>
</u></em></h2><h2><em><u>Therefore, x could be any number less than or equal to 6. In interval notation, this looks like:
</u></em></h2><h2><em><u>
</u></em></h2><h2><em><u>(
</u></em></h2><h2><em><u>−
</u></em></h2><h2><em><u>∞
</u></em></h2><h2><em><u>,
</u></em></h2><h2><em><u>6
</u></em></h2><h2><em><u>]
</u></em></h2><h2><em><u>
</u></em></h2><h2><em><u>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 </u></em></h2><h2><em><u>x
</u></em></h2><h2><em><u> be any number less than 6, but it could also be 6.
</u></em></h2><h2><em><u>
</u></em></h2><h2><em><u>Let's try the second example:
</u></em></h2><h2><em><u>
</u></em></h2><h2><em><u>3
</u></em></h2><h2><em><u>x
</u></em></h2><h2><em><u>−
</u></em></h2><h2><em><u>2
</u></em></h2><h2><em><u>4
</u></em></h2><h2><em><u>>
</u></em></h2><h2><em><u>4
</u></em></h2><h2><em><u>
</u></em></h2><h2><em><u>3
</u></em></h2><h2><em><u>x
</u></em></h2><h2><em><u>−
</u></em></h2><h2><em><u>2
</u></em></h2><h2><em><u>>
</u></em></h2><h2><em><u>16
</u></em></h2><h2><em><u>
</u></em></h2><h2><em><u>3
</u></em></h2><h2><em><u>x
</u></em></h2><h2><em><u>>
</u></em></h2><h2><em><u>18
</u></em></h2><h2><em><u>
</u></em></h2><h2><em><u>x
</u></em></h2><h2><em><u>>
</u></em></h2><h2><em><u>6
</u></em></h2><h2><em><u>
</u></em></h2><h2><em><u>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:
</u></em></h2><h2><em><u>
</u></em></h2><h2><em><u>(
</u></em></h2><h2><em><u>6
</u></em></h2><h2><em><u>,
</u></em></h2><h2><em><u>∞
</u></em></h2><h2><em><u>)
</u></em></h2><h2><em><u>
</u></em></h2><h2><em><u>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).
</u></em></h2><h2><em><u>
</u></em></h2><h2><em><u>Now, the problem used the word "OR", meaning that either of these equations could be true. That means that either </u></em></h2><h2><em><u>x
</u></em></h2><h2><em><u> is on the interval </u></em></h2><h2><em><u>(
</u></em></h2><h2><em><u>−
</u></em></h2><h2><em><u>∞
</u></em></h2><h2><em><u>,
</u></em></h2><h2><em><u>6
</u></em></h2><h2><em><u>]
</u></em></h2><h2><em><u> or the interval </u></em></h2><h2><em><u>(
</u></em></h2><h2><em><u>6
</u></em></h2><h2><em><u>,
</u></em></h2><h2><em><u>∞
</u></em></h2><h2><em><u>)
</u></em></h2><h2><em><u>. In other words, </u></em></h2><h2><em><u>x
</u></em></h2><h2><em><u> is either less than or equal to 6, or it is greater than 6. When you combine these two statements, it becomes clear that </u></em></h2><h2><em><u>x
</u></em></h2><h2><em><u> could be any real number, since no matter what number </u></em></h2><h2><em><u>x
</u></em></h2><h2><em><u> is, it will fall in one of these intervals. The interval "all real numbers" is written like this:
</u></em></h2><h2><em><u>
</u></em></h2><h2><em><u>(
</u></em></h2><h2><em><u>−
</u></em></h2><h2><em><u>∞
</u></em></h2><h2><em><u>,
</u></em></h2><h2><em><u>∞
</u></em></h2><h2><em><u>)
</u></em></h2><h2><em><u>
</u></em></h2><h2><em><u>Final Answer</u></em></h2><h2><em><u>Step-by-step explanation:</u></em></h2>
