Which Inequality Is Graphed Below

Decoding Inequalities: Identifying the Graphed Inequality

This article will guide you through the process of identifying the inequality represented by a given graph. Understanding how to interpret graphs of inequalities is crucial in algebra and beyond, forming the foundation for solving real-world problems involving constraints and relationships. We will cover various types of inequalities, including linear inequalities in one and two variables, and provide a step-by-step approach to accurately determining the inequality from its graphical representation. This will involve understanding shading, boundary lines (solid vs. dashed), and how to translate graphical information into algebraic notation.

Understanding the Components of an Inequality Graph

Before we dive into specific examples, let's familiarize ourselves with the key components you'll encounter when interpreting inequality graphs:

• Boundary Line: This is the line that separates the plane into two regions. The equation of this line is crucial in determining the inequality. It represents the equality part of the inequality (e.g., if the inequality is y > 2x + 1, the boundary line is y = 2x + 1).

• Solid vs. Dashed Line: A solid line indicates that the points on the boundary line are included in the solution set (≥ or ≤). A dashed line indicates that the points on the boundary line are not included in the solution set (< or >).

• Shaded Region: The shaded region represents the solution set of the inequality. Any point within this region satisfies the inequality.

• Coordinate Plane: The graph is typically drawn on a coordinate plane with x and y axes, allowing us to identify the coordinates of points.

Linear Inequalities in One Variable

Let's start with the simplest case: linear inequalities in one variable. These inequalities involve only one variable (usually x) and are represented on a single number line.

Example:

Imagine a graph showing a shaded region to the right of 3 on a number line, with a solid circle at 3. This represents the inequality x ≥ 3. The solid circle indicates that 3 is included in the solution set. If the circle were open (hollow), it would represent x > 3. A shaded region to the left of a point indicates values less than that point (e.g., x < -2 or x ≤ -2).

Key takeaway: In one-variable inequalities, the direction of the shading indicates whether the solution includes values greater than or less than the boundary point. The type of circle (solid or open) determines whether the boundary point itself is included.

Linear Inequalities in Two Variables

Linear inequalities in two variables (typically x and y) are represented on a coordinate plane. These are more complex but follow a systematic approach for identification.

Step-by-Step Guide to Identifying the Inequality:

1. Determine the Equation of the Boundary Line: Identify the slope and y-intercept of the boundary line (if it's not already explicitly given). Use the slope-intercept form (y = mx + b) or the point-slope form to find the equation. Remember, the boundary line represents the equality part of the inequality.

2. Identify the Type of Boundary Line: Is the line solid or dashed? A solid line indicates ≥ or ≤, while a dashed line indicates > or <.

3. Determine the Shaded Region: Observe which region of the coordinate plane is shaded. This shaded area represents the solution set of the inequality.

4. Test a Point: Choose a point in the shaded region. Substitute the coordinates of this point into the equation of the boundary line. If the inequality is true, you have the correct inequality. If it's false, you need to reverse the inequality sign (e.g., change > to <, or ≥ to ≤).

Example 1:

Let's say the graph shows a dashed line with the equation y = 2x + 1, and the region above the line is shaded.

• Step 1: The boundary line equation is y = 2x + 1.
• Step 2: The line is dashed, indicating > or <.
• Step 3: The region above the line is shaded.
• Step 4: Let's test the point (0, 2). Substituting into y = 2x + 1 gives 2 = 2(0) + 1, which simplifies to 2 = 1. This is false. Since the point (0,2) is in the shaded region, we must use the inequality that makes the statement true when substituting the point (0,2). Thus, the inequality is y > 2x + 1.

Example 2:

The graph shows a solid line with the equation y = -x + 3, and the region below the line is shaded.

• Step 1: The boundary line equation is y = -x + 3.
• Step 2: The line is solid, indicating ≥ or ≤.
• Step 3: The region below the line is shaded.
• Step 4: Let's test the point (0, 0). Substituting into y = -x + 3 gives 0 = -(0) + 3, which simplifies to 0 = 3. This is false. Therefore, we use the inequality that would make the statement true when the point (0,0) is substituted, resulting in the inequality y ≤ -x + 3.

Example 3: Handling Horizontal and Vertical Lines

Consider a graph with a shaded region to the left of a vertical line at x = 2. The line itself is solid. This represents the inequality x ≤ 2. Similarly, a shaded region above a horizontal line at y = -1 (solid line) represents y ≥ -1. Remember that horizontal lines have a slope of 0, and vertical lines have an undefined slope.

Example 4: Dealing with inequalities involving absolute values:

Absolute value inequalities create V-shaped regions. For instance, an inequality like |x| < 2 would be represented by the region between the vertical lines x = -2 and x = 2, excluding the lines themselves (dashed lines). The inequality |x| ≥ 2 would include those lines as solid, and include all regions outside the band between x = -2 and x = 2.

Non-Linear Inequalities

While the focus has been on linear inequalities, it’s important to acknowledge that inequalities can involve non-linear functions such as parabolas (quadratic inequalities), circles (circular inequalities), or more complex curves. The principles remain largely the same: identify the boundary curve, determine if it's included (solid) or excluded (dashed), and test a point within the shaded region to determine the correct inequality symbol. However, identifying the equation of the boundary curve may require more advanced algebraic techniques.

Frequently Asked Questions (FAQ)

• Q: What if I choose a point in the unshaded region when testing? A: If you choose a point in the unshaded region and find that the inequality is true, you know you have the wrong inequality symbol. Reverse the inequality symbol to get the correct inequality.

• Q: What if the inequality is in standard form (Ax + By ≤ C)? A: You can still graph it. Find the x and y intercepts and draw the line. Then test a point to determine the correct inequality sign.

• Q: Can I use a graphing calculator to check my work? A: Yes, graphing calculators can be extremely helpful in verifying your results.

Conclusion

Identifying the inequality represented by a graph is a fundamental skill in algebra. By understanding the components of an inequality graph (boundary line, shading, solid vs. dashed lines), and by following a systematic approach involving testing points, you can confidently translate graphical representations into their algebraic equivalents. Remember to pay close attention to the details – the type of line and the shaded region are critical clues in determining the correct inequality. Mastering this skill lays a strong foundation for tackling more advanced mathematical concepts and problem-solving situations. Practice with various examples, gradually increasing the complexity, will solidify your understanding and build your confidence in interpreting inequality graphs.