Which Linear Inequality Is Represented By The Graph

When examining a graph, identifying the linear inequality it represents involves understanding the relationship between the line and the shaded region. The line itself indicates the boundary, while the shading shows the region where the inequality holds true.

Understanding Linear Inequalities

A linear inequality is a mathematical statement that compares two expressions using inequality symbols such as < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). Unlike linear equations, which have one specific solution, linear inequalities have a range of solutions. A linear inequality in two variables (typically x and y) can be represented graphically on a coordinate plane.

Components of a Linear Inequality Graph

• The Boundary Line: This line represents the equation where the inequality symbol is replaced with an equal sign (=). It divides the coordinate plane into two regions. The line can be solid or dashed:
  • A solid line indicates that the points on the line are included in the solution (≤ or ≥).
  • A dashed line indicates that the points on the line are not included in the solution (< or >).
• The Shaded Region: This region represents all the points (x, y) that satisfy the inequality. The region is shaded on one side of the boundary line.
• Test Point: A test point is a point not on the boundary line, used to determine which side of the line should be shaded. If the test point satisfies the inequality, the region containing that point is shaded.

General Forms of Linear Inequalities

A linear inequality can be expressed in several forms, the most common being:

• Slope-Intercept Form: y < mx + b, y > mx + b, y ≤ mx + b, or y ≥ mx + b, where m is the slope and b is the y-intercept.
• Standard Form: Ax + By < C, Ax + By > C, Ax + By ≤ C, or Ax + By ≥ C, where A, B, and C are constants.

Steps to Determine the Linear Inequality from a Graph

1. Identify the Boundary Line:

   • Find Two Points on the Line: Look for points where the line intersects the grid lines on the coordinate plane. These points will have integer coordinates, making them easier to work with.
   • Calculate the Slope (m): Use the formula m = (y₂ - y₁) / (x₂ - x₁), where (x₁, y₁) and (x₂, y₂) are the coordinates of the two points you found.
   • Determine the y-Intercept (b): This is the point where the line crosses the y-axis. If you can't directly read it from the graph, use one of the points and the slope to solve for b in the equation y = mx + b.

2. Write the Equation of the Boundary Line:

   • Use Slope-Intercept Form: Substitute the values of m and b into the equation y = mx + b. This gives you the equation of the line.
   • Use Point-Slope Form (Alternative): If you prefer, use the point-slope form y - y₁ = m(x - x₁) with one of the points and the slope to find the equation. Then, convert it to slope-intercept form.

3. Determine the Inequality Symbol:

   • Check if the Line is Solid or Dashed:
     • Solid line: The inequality will be either ≤ or ≥.
     • Dashed line: The inequality will be either < or >.
   • Choose a Test Point: Select a point that is not on the line. The easiest point is often (0,0), if the line does not pass through the origin.
   • Substitute the Test Point into the Equation: Plug the x and y values of the test point into the equation y = mx + b.
   • Determine the Correct Inequality Symbol:
     • If the test point is in the shaded region and satisfies y > mx + b, use the > symbol (if the line is dashed) or ≥ symbol (if the line is solid).
     • If the test point is in the shaded region and satisfies y < mx + b, use the < symbol (if the line is dashed) or ≤ symbol (if the line is solid).
     • If the test point is not in the shaded region, reverse the above rules.

4. Write the Linear Inequality:

   • Combine the equation of the boundary line with the appropriate inequality symbol to write the linear inequality.

Examples

Example 1:

Graph Description: A solid line passes through the points (0, 2) and (2, 0). The region below the line is shaded.

1. Identify the Boundary Line:

   • Two points: (0, 2) and (2, 0)
   • Slope: m = (0 - 2) / (2 - 0) = -2 / 2 = -1
   • y-intercept: b = 2 (from the point (0, 2))

2. Write the Equation of the Boundary Line:

   • y = -x + 2

3. Determine the Inequality Symbol:

   • Line type: Solid line, so it will be either ≤ or ≥.
   • Test point: (0, 0)
   • Substitute (0, 0) into y = -x + 2: 0 = -0 + 2 => 0 = 2 (False)
   • Since (0, 0) is in the shaded region and makes the equation false, we need to use the opposite inequality. Thus, we use ≥.

4. Write the Linear Inequality:

   • y ≥ -x + 2

Example 2:

Graph Description: A dashed line passes through the points (-1, 0) and (0, 1). The region above the line is shaded.

1. Identify the Boundary Line:

   • Two points: (-1, 0) and (0, 1)
   • Slope: m = (1 - 0) / (0 - (-1)) = 1 / 1 = 1
   • y-intercept: b = 1 (from the point (0, 1))

2. Write the Equation of the Boundary Line:

   • y = x + 1

3. Determine the Inequality Symbol:

   • Line type: Dashed line, so it will be either < or >.
   • Test point: (0, 2) (a point clearly in the shaded region)
   • Substitute (0, 2) into y = x + 1: 2 = 0 + 1 => 2 = 1 (False)
   • Since the point (0,2) is in the shaded region and makes the equality false, we reverse the inequality symbol. Since the line is dashed, the inequality is >.

4. Write the Linear Inequality:

   • y > x + 1

Example 3:

Graph Description: A solid vertical line passes through x = 3. The region to the left of the line is shaded.

1. Identify the Boundary Line:

   • The line is vertical, so the equation is of the form x = c.
   • The line passes through x = 3, so the equation is x = 3.

2. Determine the Inequality Symbol:

   • Line type: Solid line, so it will be either ≤ or ≥.
   • The region to the left is shaded, meaning x values are less than 3. Since the line is solid, the inequality is ≤.

3. Write the Linear Inequality:

   • x ≤ 3

Example 4:

Graph Description: A dashed horizontal line passes through y = -2. The region above the line is shaded.

1. Identify the Boundary Line:

   • The line is horizontal, so the equation is of the form y = c.
   • The line passes through y = -2, so the equation is y = -2.

2. Determine the Inequality Symbol:

   • Line type: Dashed line, so it will be either < or >.
   • The region above is shaded, meaning y values are greater than -2. Since the line is dashed, the inequality is >.

3. Write the Linear Inequality:

   • y > -2

Common Pitfalls

• Forgetting to Check the Line Type: Always determine if the line is solid or dashed before choosing the inequality symbol.
• Choosing the Wrong Test Point: Make sure your test point is not on the boundary line.
• Incorrectly Calculating the Slope: Double-check your calculations, especially when dealing with negative numbers.
• Not Reversing the Inequality Symbol: If the test point does not satisfy the equation, remember to reverse the inequality symbol.
• Assuming (0,0) Always Works: If the line passes through the origin, you cannot use (0,0) as a test point. Choose another point that is clearly on one side of the line.
• Confusing Slope-Intercept and Standard Forms: Remember that while slope-intercept form (y = mx + b) is often easier to work with, the standard form (Ax + By = C) can sometimes be more convenient depending on the context.

Using Standard Form

While the examples above primarily use slope-intercept form, understanding how to work with the standard form Ax + By < C is also crucial. Here’s how to approach it:

1. Convert to Slope-Intercept Form: Solve the inequality for y to get it into the form y < mx + b. This allows you to easily identify the slope and y-intercept.

2. Analyze the Graph: As with slope-intercept form, identify the boundary line, whether it's solid or dashed, and the shaded region.

3. Choose a Test Point: Select a point not on the line and substitute its coordinates into the original standard form inequality.

4. Determine the Inequality Symbol:

   • If the test point satisfies the inequality in standard form and is in the shaded region, the original inequality symbol is correct.
   • If the test point does not satisfy the inequality and is in the shaded region, reverse the inequality symbol.

Example Using Standard Form:

Graph Description: A dashed line is represented by 2x + 3y = 6. The region above the line is shaded.

1. Convert to Slope-Intercept Form:

   • 3y = -2x + 6
   • y = (-2/3)x + 2

2. Analyze the Graph:

   • The boundary line is y = (-2/3)x + 2.
   • It's a dashed line.
   • The region above is shaded.

3. Choose a Test Point:

   • Use (0, 3), a point in the shaded region.

4. Determine the Inequality Symbol:

   • Substitute (0, 3) into the original inequality 2x + 3y > 6 (we are testing the 'greater than' because the region above is shaded and the line is dashed):

     • 2(0) + 3(3) > 6
     • 0 + 9 > 6
     • 9 > 6 (True)

   • Since the test point (0, 3) satisfies the inequality 2x + 3y > 6, the inequality symbol is correct.

5. Write the Linear Inequality:

   • 2x + 3y > 6

Special Cases

• Horizontal Lines: These are in the form y = c. The inequality will be y < c, y > c, y ≤ c, or y ≥ c. Determine the correct symbol by observing whether the shading is above or below the line and whether the line is solid or dashed.
• Vertical Lines: These are in the form x = c. The inequality will be x < c, x > c, x ≤ c, or x ≥ c. Determine the correct symbol by observing whether the shading is to the left or right of the line and whether the line is solid or dashed.

Applications of Linear Inequalities

Linear inequalities are used in various real-world applications, including:

• Budgeting: Determining how much of different items you can afford given a limited budget.
• Resource Allocation: Optimizing the use of resources like labor, materials, and equipment to maximize output.
• Business Planning: Analyzing the break-even point for a product or service.
• Nutrition: Planning a diet that meets specific nutritional requirements within certain constraints.
• Optimization Problems: In operations research, linear inequalities are used to define constraints in linear programming problems, where the goal is to maximize or minimize a linear objective function.

Advanced Techniques

• Systems of Linear Inequalities: When you have multiple inequalities graphed on the same coordinate plane, the solution is the region where all the inequalities are satisfied simultaneously. This region is the intersection of all the shaded regions.
• Linear Programming: This involves finding the optimal solution to a problem that is subject to linear constraints. Linear inequalities define the feasible region, and the optimal solution is found at one of the vertices of this region.
• Sensitivity Analysis: Analyzing how changes in the coefficients of the linear inequalities or the objective function affect the optimal solution.

Conclusion

Identifying the linear inequality represented by a graph involves a systematic approach that includes finding the equation of the boundary line, determining the correct inequality symbol, and understanding the shaded region. By following the steps outlined above and practicing with various examples, you can confidently interpret and represent linear inequalities graphically. The ability to work with linear inequalities is a valuable skill in mathematics and has wide-ranging applications in various fields, making it an essential topic to master.