Problem 113 - Introductory and Intermediate Algebra for College Students 4th

Question

When graphing the solutions of an inequality, what is the difference between a parenthesis and a bracket?

Step-by-Step Solution

Answer

In the graphical representation of inequalities, a parenthesis denotes that the endpoint is not included in the solution set. Conversely, a bracket shows that the endpoint is included.

Step 1: Introduction to Graphical Representation of Inequalities

When plotting inequalities on a number line, a parenthesis '(' or ')' is used to denote that the end value is not included in the solution interval, while a bracket '[' or ']' shows that the end value is included.

Step 2: Interpretation of a Parenthesis

The parenthesis signifies that the end value is not part of the solution. For instance, for an inequality 'x > 2', on the graph, a parenthesis would be used at 2, because '2' is not part of the solution as 'x' is only greater than '2', not greater than or equal to '2'.

Step 3: Interpretation of a Bracket

When a bracket is used, it demonstrates that the endpoint value is part of the solution. This is generally used for inequalities showing a greater than or equal to '>=', or a lesser than or equal to '<=' relationship. For example, in 'x >= 5', on a number line, '5' would be indicated with a bracket, as 'x' could also be equal to '5'.

Key Concepts

Understanding Number Line Representation

When working with inequalities, visualizing them on a number line is a helpful way to understand their solutions. A number line is simply a straight line on which numbers are placed at uniform distances.
It's a fundamental graphic tool that aids in better comprehending where the solutions to inequalities lie in relation to each other.

• To plot inequalities on a number line, you mark the solutions or intervals that satisfy the inequality.
• Sometimes only a part of the number line fulfills the inequality and other times, the solution might be a single point.
• You use different symbols, typically parentheses and brackets, to show whether endpoints are included in the solution or not.

Understanding number line representation is crucial as it simplifies the complexity of inequalities into a visual format that is easier to digest. It helps learners see the range of possible values and understand the nuances of open and closed intervals.

The Use of Parentheses in Inequalities

Parentheses in inequalities are a key way to illustrate that an endpoint value is excluded from the solution set. These symbols-- ')' and '('-- are used when the inequality is strict, meaning the variable does not include the endpoint itself.
For instance, if you have the inequality \( x > 3 \), it means that \( x \) is any number greater than 3, but not 3 itself.

• On a number line, you would use a parenthesis at 3 to show that this point is not part of the solution.
• The number line would extend infinitely to the right from this open circle, indicating all numbers greater than 3 are included.
• Parentheses indicate an open interval, meaning the boundary is approached but not reached.

Using parentheses helps make it clear exactly which values are included and which are not, thereby reducing confusion and clarifying the solution set boundaries.

Understanding Brackets in Inequalities

Brackets in inequalities serve to show that the endpoint value is included in the solution set. Represented by the symbols ']' and '[', they are used when inequalities indicate a greater than or equal to (\( \geq \)) or less than or equal to (\( \leq \)) relationship.
When you see a bracket, it means the endpoint is part of the solution.

• Consider the inequality \( x \leq 4 \). On a number line, you would use a square bracket at 4 to indicate that 4 is included in the solution.
• The line would stretch infinitely to the left from this bracket, signifying all numbers less than or equal to 4 are included.
• A bracket indicates a closed interval, showing the solution includes the boundary point.

By using brackets, the graphical representation becomes more precise, making it clear that endpoints should be counted as part of the solutions, which is vital for correctly interpreting the range of possible solutions to an inequality.