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Chapter 6: Problem 3

Decide whether you would use an open dot or a solid dot to graph the inequality. $$ a<3 $$

Short Answer

Expert verified
In graphing the inequality 'a<3', an open dot will be used since the number 3 is not part of the solution set.

Step by step solution

01

Ground Rule

To decide whether an open dot or a solid dot should be used for graphing an inequality, you need to understand the difference: a solid dot is used when a number is included in the solution set (representing 'greater than or equal to' or 'less than or equal to'), whereas an open dot is used when the number is not part of the solution set (representing 'greater than' or 'less than').
02

Evaluate the Inequality

Looking at the given inequality, 'a<3', it can be seen that 3 is not included in the inequality. It states that 'a' is less than 3 but not 'less than or equal to' 3.
03

Final Decision

So, given the nature of the inequality where the number 3 is not included, an open dot will be used to graph this inequality. If 'a' was less than or equal to 3 (a<=3), we would use a solid dot as 3 would be included in the solution set.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Open Dot
An open dot on a number line represents the concept of not including a particular value in the solution set of an inequality. This is often used when the inequality symbol is either '<' or '>' without an equality component.
For example, in the inequality \(a < 3\), the open dot is used above the number 3. It visually communicates that 3 is a boundary point, but not included in the set of possible values for \(a\).
  • Aids in distinguishing strict inequalities like \(a < 3\) or \(a > 5\).
  • Indicates that all numbers less than 3 are valid solutions in \(a < 3\).
Open dots are usually drawn as simple circles over the indicated number on a number line. When learning this concept, remember that an open dot will be your go-to choice when the endpoint is not part of the solution.
Solid Dot
A solid dot conveys that a number is included as part of the solution set for an inequality. It is used when the inequality symbol includes equality, such as '\(\leq\)' or '\(\geq\)'.
In an inequality like \(a \leq 3\), a solid dot would be placed on 3. This tells you that not only are all numbers less than 3 solutions, but 3 itself is also included.
  • Helps differentiate between inclusive inequalities \(a \leq 3\) or \(a \geq 5\) and strict inequalities.
  • Ensures clarity in graphical representation, highlighting that the endpoint is part of the solution.
When grappling with whether to use a solid dot, ask yourself if the boundary number should be part of your solution. If the answer is yes, roll with the solid dot!
Solution Set
The solution set of an inequality includes all possible values that satisfy the inequality. The graphical representation of this solution set on a number line can include either open or solid dots.
Using the inequality \(a<3\):
  • The values of \(a\) can be any number less than 3.
  • 3 itself is not part of the solutions, aligning with the use of an open dot at 3.
Visualizing the solution set is often easier with number lines as they provide a quick intuitive grasp.
When thinking about solution sets, remember:
  • Solid dots imply the endpoint is part of the solution.
  • Open dots imply the endpoint is *not* part of the solution.
Understanding whether an inequality includes or excludes its boundary number is key to identifying the correct solution set.

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