Which Graph Represents The Inequality Y 1-3x

Which Graph Represents the Inequality y ≤ 1 - 3x? A Comprehensive Guide

Understanding how to graphically represent inequalities is crucial in algebra and beyond. This comprehensive guide will walk you through the process of identifying the correct graph for the inequality y ≤ 1 - 3x. We'll explore the steps involved, common pitfalls, and provide you with the tools to confidently tackle similar problems.

Understanding Linear Inequalities

Before diving into the specific inequality, let's refresh our understanding of linear inequalities. A linear inequality is similar to a linear equation (like y = mx + b), but instead of an equals sign (=), it uses an inequality symbol: < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). These symbols indicate a range of values rather than a single point.

The general form of a linear inequality is:

Ax + By ≤ C (or using any of the other inequality symbols)

Graphing Linear Inequalities: A Step-by-Step Approach

Graphing linear inequalities involves several key steps:

1. Rewrite the Inequality in Slope-Intercept Form: The easiest way to graph a linear inequality is to rewrite it in slope-intercept form: y = mx + b, where 'm' is the slope and 'b' is the y-intercept.

   For our inequality, y ≤ 1 - 3x, we already have it in a very close form. Rearranging slightly, we get:

   y ≤ -3x + 1

2. Identify the Slope and Y-intercept: From the slope-intercept form, we can easily identify the slope and y-intercept:

   • Slope (m) = -3: This means that for every 1 unit increase in x, y decreases by 3 units.
   • Y-intercept (b) = 1: This is the point where the line crosses the y-axis (at (0, 1)).

3. Graph the Boundary Line: Now, graph the line y = -3x + 1. This is the boundary line of our inequality. Because the inequality includes "≤" (less than or equal to), the boundary line will be a solid line, indicating that points on the line are included in the solution set. If the inequality was "<" or ">", the boundary line would be a dashed line.

4. Shade the Appropriate Region: This is the most crucial step. The inequality y ≤ -3x + 1 means that we are looking for all the points where the y-coordinate is less than or equal to -3x + 1. To determine which side of the boundary line to shade, choose a test point that is not on the line. A convenient test point is usually the origin (0, 0).

   Let's substitute (0, 0) into the inequality:

   0 ≤ -3(0) + 1 0 ≤ 1

   This statement is true. Since the test point (0, 0) satisfies the inequality, we shade the region that contains the origin. If the statement were false, we would shade the region on the opposite side of the line.

Identifying the Correct Graph: Key Features to Look For

When presented with multiple graphs, here's how to identify the one representing y ≤ -3x + 1:

• Solid Line: The graph must have a solid line, not a dashed line, because of the "≤" symbol.
• Y-intercept at (0, 1): The line must cross the y-axis at the point (0, 1).
• Negative Slope: The line must slope downwards from left to right because the slope is -3.
• Shading Below the Line: The region below the line (the area containing the origin) must be shaded.

Common Mistakes to Avoid

• Confusing Dashed and Solid Lines: Remember, a dashed line represents strict inequalities (< or >), while a solid line represents inequalities that include equality (≤ or ≥).
• Shading the Wrong Region: Always test a point to determine the correct region to shade. Failing to do this is a common error.
• Incorrect Slope or Y-intercept: Double-check your calculations when determining the slope and y-intercept to avoid errors in graphing the line.
• Neglecting the Inequality Symbol: The inequality symbol dictates whether the line is solid or dashed and which region to shade. Carefully consider its meaning.

Extending the Understanding: Variations and Applications

The principles discussed above apply to all linear inequalities. Let's look at a few variations:

• y ≥ 1 - 3x: This would be graphed similarly, but the shaded region would be above the line, and the line itself would still be solid.
• y > 1 - 3x: This would use a dashed line, indicating that points on the line are not included in the solution set. The shaded region would still be above the line.
• y < 1 - 3x: This would also use a dashed line, and the shaded region would be below the line.

Linear inequalities have numerous applications in various fields, including:

• Optimization Problems: In operations research and linear programming, inequalities are used to model constraints and find optimal solutions.
• Economics: Inequalities are used to represent budget constraints, resource limitations, and other economic relationships.
• Computer Graphics: Inequalities are used to define regions and shapes in computer graphics.
• Engineering: Inequalities are used to model constraints and tolerances in engineering design.

Conclusion: Mastering Linear Inequalities

Graphing linear inequalities might seem challenging at first, but with a systematic approach and a clear understanding of the concepts, you can master this skill. Remember the key steps: rewrite in slope-intercept form, identify the slope and y-intercept, graph the boundary line (solid or dashed), and shade the appropriate region using a test point. By carefully following these steps and avoiding common mistakes, you will confidently identify the correct graph for any linear inequality, including y ≤ 1 - 3x. Practice is key to building proficiency and ensuring accuracy in your graphical representations. Through consistent practice and attention to detail, you'll become adept at visualizing and interpreting linear inequalities, unlocking their practical applications in diverse fields.