Question

Tell whether you should use an open dot or a closed dot on the graph of the inequality. \(5 x-3 \geq 12 \quad\)

Short answer

In the graph of the inequality \(5x - 3 \geq 12\), a closed dot should be used.

Step-by-step solution

Understand the inequality

The inequality given is \(5x - 3 \geq 12\). Here, the inequality used is 'greater than or equal to', represented by '≥'.

Dot determination based on inequality type

In inequality graphing, an open dot is used for '<' and '>' to indicate that the endpoint is not included in the solution. A closed dot is used for '≤' and '≥' to show that the endpoint is included in the solution.

Apply the rule to the given inequality

In our inequality \(5x - 3 \geq 12\), a 'greater than or equal to' sign is used. This means that the endpoint is included in the solution. Therefore, it should be represented with a closed dot on the graph.

Key concepts

Open and Closed Dots

Graphing inequalities on a number line involves marking a point to show the boundary of the solution set. Crucially, how we mark this point—with an open dot or a closed dot—provides information about whether the boundary number is part of the solution.

An open dot indicates that the boundary number itself is not included in the solution. It's used when the inequality is either '<' (less than) or '>' (greater than). On the other hand, a closed dot is used when the inequality includes the boundary number, indicating a '≤' (less than or equal to) or '≥' (greater than or equal to) relationship.

For example, if you come across the inequality x < 3, you'd place an open dot at 3 on the number line to signify that 3 is not a solution to the inequality. Meanwhile, for x ≤ 3, a closed dot at 3 communicates that 3 is a valid solution.

Inequality Symbols

Understanding inequality symbols is essential for both solving and graphing inequalities. There are four primary symbols to be aware of: '<' (less than), '>' (greater than), '≤' (less than or equal to), and '≥' (greater than or equal to).

The '<' and '>' symbols signify that the values on one side of the inequality are either strictly less than or strictly greater than those on the other side. In contrast, the '≤' and '≥' symbols include equality, indicating that the values could also be equal. This small, yet significant, inclusion of the equal sign changes how we approach the solution and graphing process.

It's important to accurately interpret these symbols since they determine the solution set and how it's represented graphically, contributing to a correct understanding of the inequality.

Solution of Inequalities

The solution of inequalities consists of all the values that make the inequality true. When we solve an inequality, we're looking for a range of numbers rather than a single value, as is the case with equations. Graphing is a powerful way to visualize this set of solutions.

To represent the solution graphically, we determine the critical value(s), decide whether to use an open or closed dot based on the inequality symbol, and then shade the portion of the number line where the inequality holds true. For example, if the solution to an inequality is all numbers less than 5, we place an open dot at 5 and shade everything to the left of that dot.

Remember that inequalities can also have multiple ranges as their solution, such as with compound inequalities. The graphing method, therefore, provides a clear and immediate visual representation of all possible solutions, making it an invaluable tool in understanding and solving inequalities.