Please help with the following:
What is an open dot, and what is a closed dot?
Also, when are they used in inequalities?
I am not really sure about what they are. The lesson has not taught me anything about open dots and closed dots. I have also looked them up, but I got confused. Please help, and it will be greatly appreciated. :)
a closed dot means included
like x < 5 means less than (open)
x </= 5 means less than or equal to (closed)
http://mathcentral.uregina.ca/QQ/database/QQ.09.05/cynthia1.html
Oh, now I get it!
Less than(<) and greater than(>) are open, while less than or equal to(≤) and greater than or equal to(≥) are closed.
Thank you, Damon! Your help is again, greatly appreciated. :)
You are welcome :)
An open dot (○) and a closed dot (●) are used to represent intervals or solutions in inequalities. They are primarily used in number line graphs to illustrate whether a specific value is included or excluded in the set of solutions.
To understand open and closed dots, let's start by considering a simple inequality, such as x > 2. This means that x is greater than 2. To represent this inequality on a number line, we can start by drawing a line and placing a closed dot at the number 2. A closed dot is used when the given value is included in the solution set, in this case, because x can be equal to 2.
Next, we need to show that x is greater than 2. We can use an arrow pointing to the right side of the number line to indicate that the values of x extend beyond the closed dot. The arrow means all the values greater than 2 are part of the solution.
On the other hand, if the inequality is x ≥ 2, which means x is greater than or equal to 2, we would use a closed dot at 2, just like before, because 2 is included in the solution set. The arrow would still point to the right, indicating all the values greater than or equal to 2.
However, if the inequality is x > 2, without the equal sign, we would use an open dot at 2, indicating that 2 is not included in the solution set. The arrow would still point to the right, including all values greater than 2 but not 2 itself.
In summary,
- A closed dot (●) is used to represent a value that is included in the solution set.
- An open dot (○) is used to represent a value that is excluded from the solution set.
- Closed dots are used when there is an inequality with an equal sign (≥, ≤) included.
- Open dots are used when there is an inequality without an equal sign (> or <) included.
Remember, these symbols are used to represent intervals and solutions graphically, especially on number lines, to indicate whether the boundary value is included or excluded in the set.