Problem 8 - Algebra for College Students

Question

Express the given inequality in interval notation and sketch a graph of the interval. \(x \leq 0\)

Step-by-Step Solution

Answer

Interval: \((-\infty, 0]\). Solid line shaded to the left of 0, solid circle at 0.

Step 1: Understand the Inequality

The given inequality is \(x \leq 0\). This means that \(x\) includes all values less than or equal to zero. So, \(0\) is included in the set as well as all negative numbers.

Step 2: Express in Interval Notation

To express \(x \leq 0\) in interval notation, we start from minus infinity (\(-\infty\)) and go up to zero (0), including zero. In interval notation, this is expressed as \((-\infty, 0]\). We use a parenthesis for minus infinity because infinity is never included in the set, and a bracket for zero to indicate it is included.

Step 3: Sketch the Graph

Draw a number line. - Mark a point at 0 and shade all values to the left of 0. - Place a filled (solid) circle on 0 to show that zero is included in the interval. - Extend the shading endlessly to the left to indicate all numbers in this direction are included.

Key Concepts

Interval Notation

Interval notation is a concise way of writing subsets of the real number line. It is especially useful when describing solutions to inequalities. For the inequality \(x \leq 0\), interval notation helps us communicate the solution set in a neat form. When using interval notation:

• Parentheses \(( )\) indicate that an endpoint is not included in the interval. This happens with infinity (\

Number Line Representation

Representing inequalities on a number line visually demonstrates the range of possible solutions. The inequality \(x \leq 0\) indicates that values can be zero or any number less than zero. When displaying this on a number line:

• Draw a horizontal line to represent the number scale.
• Mark the point for zero clearly on the line.
• Use a filled circle at zero to show it's a part of the solution set.
• Shade the line extending leftwards from zero to depict all numbers less than zero are included.

The direction of shading is crucial in showing which numbers are solutions. Always remember:- Filled circles mean the endpoint is included.- Open circles mean the endpoint is not included.

Graphical Solutions of Inequalities

Graphical representation is a powerful tool to simplify understanding of inequalities. For a specific inequality like \(x \leq 0\), graphing it provides a visual depiction of the solution space. Graphical solutions involve the following:

• Using the number line as the base for plotting.
• Identifying critical points such as zero in this case, where the inequality transitions.
• Including symbols (like filled circles) to denote whether particular points are included in the solution.
• Highlighting shaded segments to show all numbers that make the inequality true.

Through graphical solutions, abstract inequalities become more tangible, improving comprehension and accuracy in interpreting the range of applicable values.