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

Question

Graph the solutions of each inequality on a number line. $$x<0$$

Step-by-Step Solution

Answer

The solutions to the inequality \(x < 0\) are all real numbers less than 0. This is shown on the number line as a line which extends leftward from 0, excluding 0.

Step 1: Write down the inequality

In our case, the given inequality is \(x < 0\). This means all values of \(x\) must be less than 0 for the inequality to hold.

Step 2: Identify the initial point for the graph

The inequality indicates that our interval extends to the left, starting from 0 (which is not included) until negative infinity. Therefore, 0 will be the starting point for our graph on the number line.

Step 3: Draw the number line

A number line is required to graphically represent the solution. Draw a line, label 0 at any point on the line, label negative numbers to the left of 0 and positive numbers to the right of 0.

Step 4: Graph the inequality on the number line

Since \(x < 0\), the part of number line containing values \(x\) less than 0 must be shaded. Furthermore, since \(x\) cannot be equal to 0, a parethesis instead of a solid dot should be marked at 0 to show that 0 is excluded.

Key Concepts

Understanding the Number Line

A number line is a simple and effective tool for visualizing numbers and solving mathematical problems such as inequalities. It is a straight, horizontal line that is marked with evenly spaced numbers. Here, zero usually occupies a central position, with negative numbers extending to the left and positive numbers to the right.

Key features of a number line include:

• Clear marking of numbers: Each point on the line corresponds to a real number.
• Infinite nature: Although we typically only draw a portion of it, a number line extends infinitely in both directions.
• Relative positioning: It clearly shows the relative size of numbers, making it easier to compare them.

In graphing inequalities, the number line becomes crucial. It allows you to visually represent the set of values that satisfy the inequality. Drawn using a pencil or a digital tool, it helps map these solutions in a way that is clear and easy to interpret.

Solutions to Inequalities

Inequalities describe a relationship between two expressions that are not equal, often involving symbols like ">, <, \geq, \leq". When tasked with solving these, your aim is to find all possible values of the variable that make the inequality true.

For example, with the inequality \(x < 0\), you are finding all numbers that fall to the left of zero on the number line. That is because these numbers are less than zero and satisfy the inequality. Unlike equations, which have specific solutions, inequalities often have a range of possible solutions.

• Open Interval: When the inequality is strict (like \(<\) or \(>\)), the solution set is indicated by an open interval. Here, the boundary value (like zero in our example) is not included in the solution set.
• Closed Interval: If the inequality includes equality (using symbols like \(\geq\) or \(\leq\)), the intervals are closed, meaning the boundary value is included.

Representing these solutions graphically helps facilitate a better understanding of where these values are located on a number line.

Graphing Inequalities Mathematically

Mathematical graphing involves plotting numbers or variables on a defined axis to visualize relationships. When graphing inequalities on a number line, it's essential to use specific symbols to show whether endpoints are included or excluded.

Here are the steps to graphing a simple inequality like \(x<0\) on a number line:

• Draw the number line: Label the line with zero, and place negative numbers to the left.
• Identify the boundary point: In \(x<0\), zero is the boundary, not included in the solution. Represent it with a parenthesis or open circle at zero to indicate exclusion.
• Shade or indicate direction: Since the solution includes all numbers less than zero, shade the section of the line extending to the left from zero. This visual cue clearly indicates where the solution exists on the number line.

Graphing this way helps in interpreting inequalities effectively and lays the groundwork for understanding more complex relations and equations in mathematics.