Problem 68 - College Algebra and Calculus: An Applied Approach

Question

Writing Explain the difference between the graphs of the inequality \(x \leq 4\) on the real number line and on the rectangular coordinate system.

Step-by-Step Solution

Answer

The graph of \(x \leq 4\) on the real number line is a shaded area stretching from 4 to the left, and on the rectangular coordinate system, it's represented as a solid vertical line along \(x = 4\) with the area to the left of that line shaded.

Step 1 - Graph on Real Number Line

Plot the number 4 on the real number line. Since \(x\) should be equal to or less than 4, shade the part of the number line to the left of 4. The shaded part will start at 4 (including 4) and extend to the left side, including all the numbers less than 4.

Step 2 - Graph on Rectangular Coordinate System

When it comes to the rectangular coordinate system, an inequality like \(x \leq 4\) is translated as a vertical line drawn through \(x = 4\). As the inequality has equal to or less than symbol (\(\leq\) ), the line should be a solid one, indicating that points on the line are included in the solutions. After the vertical line is drawn, shade the area to the left of this line. This shaded region is the set of points \((x, y)\) where \(x\) is less than or equal to 4.

Key Concepts

Real Number Line

The real number line is a simple yet powerful way to visualize numbers. To understand an inequality like \(x \leq 4\), we first identify the position of 4 on the line. In this context, the line represents all possible real numbers ranging from negative infinity to positive infinity.

When dealing with inequalities, our main task is to visually differentiate the numbers that satisfy the condition. For \(x \leq 4\), this means shading the part of the line that includes 4 and extends to the left, covering every number less than 4. The process involves:

• Placing a closed dot on the number 4 to show it is included.
• Drawing a shaded line segment continuing to the left from 4, moving towards negative infinity.

This visual representation makes it clear which numbers satisfy \(x \leq 4\). The closed dot and shading help students easily interpret that 4 and all smaller numbers are solutions in this inequality.

Rectangular Coordinate System

The rectangular coordinate system, also known as the Cartesian plane, adds another dimension to graphing, allowing for a two-variable representation, usually \(x\) and \(y\). When we graph the inequality \(x \leq 4\), the task transforms from a single line to a planar region.

Here’s how you would graph \(x \leq 4\) on this grid:

• First, draw a vertical line at \(x = 4\). This line is solid because the inequality includes values where \(x\) equals 4.
• The region on the grid that satisfies the inequality lies to the left of this solid line.

Every point in this shaded region, regardless of its \(y\)-value, satisfies the inequality. By comparing it to a simple line graph, the coordinate system offers a more comprehensive view of all the possible solutions by considering two variables.

Shading Regions

Shading regions correctly is crucial as it communicates which part of a graph satisfies a given inequality. In both one-dimensional and two-dimensional settings, shading is used to highlight the solutions.

On the real number line, shading indicates the direction of inequality along a one-dimensional space. When you shade for \(x \leq 4\), you’re marking the infinite stretch of numbers moving left towards smaller values from 4. In contrast:

• In the rectangular coordinate system, shading represents a two-dimensional area. For \(x \leq 4\), the entire left-hand side of the \(x = 4\) line is shaded, extending infinitely in both upward and downward directions along the \(y\)-axis.

It's essential to use shading rules consistently:

• A solid line or point when the inequality includes equality (\(\leq\) or \(\geq\)).
• A dashed line for strict inequalities (\( < \) or \( > \)).

Understanding shading helps interpret where the range of solutions lies and ensures an accurate depiction of inequalities on graphs, whether on a number line or a coordinate system.