Which Inequality Is Represented In The Graph Below

I am sorry, I cannot access external websites or specific files online, including images. Therefore, I am unable to determine which inequality is represented in a graph you are referencing.

However, I can provide you with a comprehensive explanation of how to identify inequalities represented in graphs and discuss various types of inequalities that commonly appear. This guide will help you understand the concepts and apply them when you have the graph in front of you.

Understanding Inequalities and Their Graphical Representation

An inequality is a mathematical statement that compares two expressions using inequality symbols. Unlike equations, which state that two expressions are equal, inequalities indicate that one expression is greater than, less than, greater than or equal to, or less than or equal to another expression. The ability to visually represent these relationships on a graph is fundamental to understanding and solving inequalities.

Basic Inequality Symbols

Before delving into graphical representations, let’s clarify the basic inequality symbols:

• > Greater than
• < Less than
• ≥ Greater than or equal to
• ≤ Less than or equal to

Representing Inequalities on a Number Line

The simplest way to visualize inequalities is on a number line. A number line is a one-dimensional graph that represents all real numbers. Here’s how to represent different types of inequalities on a number line:

• x > a: This inequality represents all numbers x that are greater than a. On a number line, this is shown by an open circle at a and a line extending to the right, indicating that all numbers to the right of a are included. The open circle signifies that a itself is not included in the solution.
• x < a: This inequality represents all numbers x that are less than a. On a number line, this is shown by an open circle at a and a line extending to the left, indicating that all numbers to the left of a are included. The open circle signifies that a itself is not included in the solution.
• x ≥ a: This inequality represents all numbers x that are greater than or equal to a. On a number line, this is shown by a closed circle at a and a line extending to the right, indicating that all numbers to the right of a are included. The closed circle signifies that a is included in the solution.
• x ≤ a: This inequality represents all numbers x that are less than or equal to a. On a number line, this is shown by a closed circle at a and a line extending to the left, indicating that all numbers to the left of a are included. The closed circle signifies that a is included in the solution.

Representing Inequalities on a Coordinate Plane

Inequalities can also be represented on a coordinate plane (the xy-plane), which is essential for visualizing inequalities with two variables. The process involves graphing a boundary line and shading the region that satisfies the inequality.

Steps to Graphing Inequalities on a Coordinate Plane

1. Replace the Inequality Symbol with an Equality Symbol: Begin by treating the inequality as if it were an equation. For example, if you have y > 2x + 1, temporarily consider it as y = 2x + 1.

2. Graph the Boundary Line: Graph the equation you obtained in step 1. This line will divide the coordinate plane into two regions. Whether the line is solid or dashed depends on the inequality symbol:

   • Solid Line: Use a solid line if the inequality includes "or equal to" (≥ or ≤). This indicates that the points on the line are part of the solution.
   • Dashed Line: Use a dashed line if the inequality does not include "or equal to" (> or <). This indicates that the points on the line are not part of the solution.

3. Choose a Test Point: Select a point that is not on the boundary line. A common choice is the origin (0, 0), if the line does not pass through it.

4. Substitute the Test Point into the Original Inequality: Plug the coordinates of the test point into the original inequality and check if the inequality holds true.

5. Shade the Appropriate Region:

   • If the test point satisfies the inequality, shade the region of the coordinate plane that contains the test point.
   • If the test point does not satisfy the inequality, shade the region of the coordinate plane that does not contain the test point.

Examples of Graphing Linear Inequalities

1. Graphing y > 2x + 1

   • Boundary Line: Graph the line y = 2x + 1. This is a linear equation with a slope of 2 and a y-intercept of 1.
   • Type of Line: Since the inequality is y > 2x + 1, use a dashed line to indicate that the points on the line are not included in the solution.
   • Test Point: Choose the origin (0, 0). Substitute into the inequality: 0 > 2(0) + 1, which simplifies to 0 > 1. This is false.
   • Shading: Since the test point (0, 0) does not satisfy the inequality, shade the region above the dashed line. This shaded region represents all points (x, y) that satisfy the inequality y > 2x + 1.

2. Graphing y ≤ -x + 3

   • Boundary Line: Graph the line y = -x + 3. This is a linear equation with a slope of -1 and a y-intercept of 3.
   • Type of Line: Since the inequality is y ≤ -x + 3, use a solid line to indicate that the points on the line are included in the solution.
   • Test Point: Choose the origin (0, 0). Substitute into the inequality: 0 ≤ -(0) + 3, which simplifies to 0 ≤ 3. This is true.
   • Shading: Since the test point (0, 0) satisfies the inequality, shade the region below the solid line. This shaded region represents all points (x, y) that satisfy the inequality y ≤ -x + 3.

Systems of Linear Inequalities

A system of linear inequalities consists of two or more inequalities considered together. The solution to a system of inequalities is the region of the coordinate plane that satisfies all the inequalities simultaneously.

Steps to Graphing Systems of Linear Inequalities

1. Graph Each Inequality: Graph each inequality separately, following the steps outlined above (graph the boundary line, determine if it is solid or dashed, choose a test point, and shade the appropriate region).

2. Identify the Region of Overlap: The solution to the system is the region where the shaded areas of all the inequalities overlap. This region represents all points (x, y) that satisfy all the inequalities in the system.

Example of Graphing a System of Linear Inequalities

Consider the following system of inequalities:

• y > x - 2
• y ≤ -2x + 4

1. Graph y > x - 2

   • Boundary Line: Graph the line y = x - 2.
   • Type of Line: Use a dashed line.
   • Test Point: Choose (0, 0). Substitute: 0 > 0 - 2, which simplifies to 0 > -2. This is true.
   • Shading: Shade the region above the dashed line.

2. Graph y ≤ -2x + 4

   • Boundary Line: Graph the line y = -2x + 4.
   • Type of Line: Use a solid line.
   • Test Point: Choose (0, 0). Substitute: 0 ≤ -2(0) + 4, which simplifies to 0 ≤ 4. This is true.
   • Shading: Shade the region below the solid line.

3. Identify the Region of Overlap: The solution to the system is the region where the shaded areas of both inequalities overlap. This region is bounded by the dashed line y = x - 2 and the solid line y = -2x + 4.

Non-Linear Inequalities

Inequalities can also involve non-linear expressions, such as quadratic, exponential, or logarithmic functions. The process of graphing these inequalities is similar to that of linear inequalities, but the boundary lines are curves instead of straight lines.

Examples of Non-Linear Inequalities

1. Graphing y > x² - 4

   • Boundary Curve: Graph the parabola y = x² - 4. This is a parabola with its vertex at (0, -4) opening upwards.
   • Type of Curve: Use a dashed line since the inequality is y > x² - 4.
   • Test Point: Choose (0, 0). Substitute: 0 > 0² - 4, which simplifies to 0 > -4. This is true.
   • Shading: Shade the region above the dashed parabola.

2. Graphing x² + y² ≤ 9

   • Boundary Curve: Graph the circle x² + y² = 9. This is a circle centered at the origin with a radius of 3.
   • Type of Curve: Use a solid line since the inequality is x² + y² ≤ 9.
   • Test Point: Choose (0, 0). Substitute: 0² + 0² ≤ 9, which simplifies to 0 ≤ 9. This is true.
   • Shading: Shade the region inside the solid circle.

Absolute Value Inequalities

Absolute value inequalities involve expressions with absolute value symbols. Recall that the absolute value of a number is its distance from zero on the number line. Solving absolute value inequalities requires considering two cases.

General Forms of Absolute Value Inequalities

1. |x| < a: This inequality means that the distance of x from zero is less than a. The solution is -a < x < a.

2. |x| > a: This inequality means that the distance of x from zero is greater than a. The solution is x < -a or x > a.

3. |x| ≤ a: This inequality means that the distance of x from zero is less than or equal to a. The solution is -a ≤ x ≤ a.

4. |x| ≥ a: This inequality means that the distance of x from zero is greater than or equal to a. The solution is x ≤ -a or x ≥ a.

Examples of Solving and Graphing Absolute Value Inequalities

1. Solve and Graph |x - 2| < 3

   • This inequality means that the distance of x - 2 from zero is less than 3.
   • Rewrite as two inequalities: -3 < x - 2 < 3.
   • Add 2 to all parts: -3 + 2 < x < 3 + 2, which simplifies to -1 < x < 5.
   • Graph: On a number line, use open circles at -1 and 5, and shade the region between them.

2. Solve and Graph |2x + 1| ≥ 5

   • This inequality means that the distance of 2x + 1 from zero is greater than or equal to 5.

   • Rewrite as two inequalities: 2x + 1 ≤ -5 or 2x + 1 ≥ 5.

   • Solve each inequality:

     • 2x + 1 ≤ -5 => 2x ≤ -6 => x ≤ -3
     • 2x + 1 ≥ 5 => 2x ≥ 4 => x ≥ 2

   • Graph: On a number line, use closed circles at -3 and 2, and shade the regions to the left of -3 and to the right of 2.

Tips for Identifying Inequalities from Graphs

1. Identify the Boundary Line or Curve: Determine the equation of the boundary line or curve. Look for intercepts, slopes, and other key features that define the equation.

2. Determine if the Line is Solid or Dashed: A solid line indicates that the points on the line are included in the solution (≥ or ≤), while a dashed line indicates that the points on the line are not included (> or <).

3. Identify the Shaded Region: The shaded region represents the set of points that satisfy the inequality. Use a test point to confirm whether the shaded region is correct.

4. Consider the Context: If the graph represents a real-world situation, consider the context of the problem to determine which inequality is appropriate. For example, if the graph represents the constraints on a budget, the inequalities might represent spending limits or minimum requirements.

Common Mistakes to Avoid

• Forgetting to Flip the Inequality Sign: When multiplying or dividing both sides of an inequality by a negative number, remember to flip the inequality sign.
• Using the Wrong Type of Line: Using a solid line when it should be dashed, or vice versa, can lead to incorrect solutions.
• Shading the Wrong Region: Always use a test point to verify that you are shading the correct region.
• Misinterpreting Absolute Value Inequalities: Remember to consider both cases when solving absolute value inequalities.

By understanding these principles and practicing with examples, you can confidently identify inequalities represented in graphs. When you have a specific graph in front of you, apply these steps to analyze the boundary line, shading, and other key features to determine the corresponding inequality. Good luck!