Lesson 7.3 Linear Inequalities In Two Variables Answer Key

Linear inequalities in two variables represent a fascinating intersection of algebra and geometry, allowing us to model real-world constraints and visualize solution sets in a coordinate plane. Understanding these inequalities is crucial for various applications, from optimization problems in economics to resource allocation in engineering. This comprehensive guide will delve into the intricacies of linear inequalities in two variables, providing a step-by-step approach to solving them and interpreting their solutions.

Understanding Linear Inequalities in Two Variables

A linear inequality in two variables, typically x and y, 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). These inequalities define a region on the coordinate plane where all points (x, y) satisfy the given inequality. Unlike linear equations, which represent a single line, linear inequalities represent a half-plane, either above or below the line, depending on the inequality symbol.

General Form:

The general form of a linear inequality in two variables is:

• Ax + By < C
• Ax + By > C
• Ax + By ≤ C
• Ax + By ≥ C

Where A, B, and C are constants, and A and B are not both zero.

Key Concepts:

• Solution Set: The set of all ordered pairs (x, y) that satisfy the inequality. This is represented graphically as a shaded region on the coordinate plane.
• Boundary Line: The line represented by the equation Ax + By = C. This line separates the coordinate plane into two half-planes.
• Solid vs. Dashed Line: If the inequality includes ≤ or ≥, the boundary line is solid, indicating that points on the line are included in the solution set. If the inequality includes < or >, the boundary line is dashed, indicating that points on the line are not included in the solution set.
• Test Point: A point not on the boundary line used to determine which half-plane to shade. If the test point satisfies the inequality, shade the half-plane containing the test point; otherwise, shade the other half-plane.

Steps to Solve and Graph Linear Inequalities in Two Variables

Solving and graphing linear inequalities involves a systematic approach to ensure accuracy and understanding. Here's a step-by-step guide:

1. Rewrite the Inequality in Slope-Intercept Form (Optional but Recommended):

This step isn't strictly necessary, but it makes graphing the boundary line easier. Solve the inequality for y to get it in the 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. Remember to reverse the inequality sign if you multiply or divide by a negative number.

Example:

Consider the inequality 2x + 3y > 6. To rewrite it in slope-intercept form, follow these steps:

• Subtract 2x from both sides: 3y > -2x + 6
• Divide both sides by 3: y > (-2/3)x + 2

2. Graph the Boundary Line:

Treat the inequality as an equation (Ax + By = C or y = mx + b) and graph the line.

• If the inequality is ≤ or ≥, draw a solid line. This indicates that the points on the line are part of the solution.
• If the inequality is < or >, draw a dashed line. This indicates that the points on the line are not part of the solution.

Example:

For the inequality y > (-2/3)x + 2, the boundary line is y = (-2/3)x + 2. Since the inequality is >, we draw a dashed line.

3. Choose a Test Point:

Select any point that is not on the boundary line. The easiest point to use is often the origin (0, 0), unless the line passes through the origin.

Example:

For the inequality y > (-2/3)x + 2, let's use the test point (0, 0).

4. Substitute the Test Point into the Inequality:

Plug the x and y coordinates of the test point into the original inequality.

Example:

Substituting (0, 0) into y > (-2/3)x + 2, we get:

0 > (-2/3)(0) + 2

0 > 2

5. Determine if the Inequality is True or False:

If the inequality is true, the test point is in the solution region. If the inequality is false, the test point is not in the solution region.

Example:

Since 0 > 2 is false, the test point (0, 0) is not in the solution region.

6. Shade the Appropriate Half-Plane:

• If the test point satisfies the inequality (the inequality is true), shade the half-plane that contains the test point.
• If the test point does not satisfy the inequality (the inequality is false), shade the half-plane that does not contain the test point.

Example:

Since (0, 0) did not satisfy the inequality y > (-2/3)x + 2, we shade the half-plane above the dashed line. This shaded region represents all the points (x, y) that satisfy the inequality.

7. Verify Your Solution:

Choose a point within the shaded region and substitute it into the original inequality. If the inequality holds true, your shading is likely correct.

Examples of Solving Linear Inequalities in Two Variables

Let's work through some examples to solidify the process:

Example 1:

Graph the inequality x + y ≤ 4

• Rewrite in slope-intercept form: y ≤ -x + 4
• Graph the boundary line: The boundary line is y = -x + 4. Since the inequality is ≤, draw a solid line. The y-intercept is 4, and the slope is -1.
• Choose a test point: Let's use (0, 0).
• Substitute the test point: 0 ≤ -0 + 4 => 0 ≤ 4
• Determine if the inequality is true or false: 0 ≤ 4 is true.
• Shade the appropriate half-plane: Shade the half-plane that contains (0, 0), which is the region below the line.

Example 2:

Graph the inequality 3x - 2y < 6

• Rewrite in slope-intercept form:
  • -2y < -3x + 6
  • y > (3/2)x - 3 (Remember to reverse the inequality sign when dividing by a negative number)
• Graph the boundary line: The boundary line is y = (3/2)x - 3. Since the inequality is >, draw a dashed line. The y-intercept is -3, and the slope is 3/2.
• Choose a test point: Let's use (0, 0).
• Substitute the test point: 0 > (3/2)(0) - 3 => 0 > -3
• Determine if the inequality is true or false: 0 > -3 is true.
• Shade the appropriate half-plane: Shade the half-plane that contains (0, 0), which is the region above the line.

Example 3:

Graph the inequality y ≥ 2x

• The inequality is already in a form that makes it easy to graph.
• Graph the boundary line: The boundary line is y = 2x. Since the inequality is ≥, draw a solid line. The line passes through the origin (0, 0) and has a slope of 2.
• Choose a test point: Since the line passes through the origin, we cannot use (0, 0) as a test point. Let's use (1, 0).
• Substitute the test point: 0 ≥ 2(1) => 0 ≥ 2
• Determine if the inequality is true or false: 0 ≥ 2 is false.
• Shade the appropriate half-plane: Shade the half-plane that does not contain (1, 0), which is the region above the line.

Applications of Linear Inequalities in Two Variables

Linear inequalities are not just abstract mathematical concepts; they have numerous real-world applications:

• Resource Allocation: Businesses use linear inequalities to model constraints on resources such as labor, materials, and time. They can then use linear programming techniques (which build upon linear inequalities) to optimize production and minimize costs.
• Diet Planning: Dietitians use linear inequalities to design meal plans that meet specific nutritional requirements, such as minimum and maximum levels of vitamins, minerals, and calories.
• Manufacturing: Manufacturers use linear inequalities to define acceptable ranges for product dimensions and quality control parameters.
• Engineering: Engineers use linear inequalities to design structures and systems that meet safety and performance criteria. For example, they might use inequalities to ensure that a bridge can withstand a certain load or that a circuit can handle a certain voltage.
• Finance: Financial analysts use linear inequalities to model investment constraints and risk tolerance levels.
• Optimization Problems: Linear inequalities form the foundation of linear programming, a powerful optimization technique used to find the best possible solution to a problem subject to certain constraints.

Solving Systems of Linear Inequalities

A system of linear inequalities consists of two or more linear inequalities that are considered simultaneously. The solution to a system of linear inequalities is the set of all points (x, y) that satisfy all of the inequalities in the system. Graphically, the solution is the region where the shaded regions of all the inequalities overlap.

Steps to Solve a System of Linear Inequalities:

1. Graph each inequality individually: Follow the steps outlined above to graph each linear inequality in the system. Be sure to use different colors or shading patterns for each inequality to distinguish them.
2. Identify the overlapping region: The solution to the system is the region where all the shaded regions overlap. This region is sometimes called the feasible region.
3. Determine the vertices of the feasible region: The vertices of the feasible region are the points where the boundary lines of the inequalities intersect. These points are important for optimization problems. You can find the coordinates of the vertices by solving the system of equations formed by the intersecting lines.
4. The feasible region can be bounded or unbounded: A bounded feasible region is completely enclosed, while an unbounded feasible region extends infinitely in one or more directions.

Example:

Solve the following system of linear inequalities:

• x + y ≤ 5
• 2x - y ≥ 0

1. Graph each inequality:

   • x + y ≤ 5 => y ≤ -x + 5 (Solid line, shade below)
   • 2x - y ≥ 0 => y ≤ 2x (Solid line, shade below)

2. Identify the overlapping region: The overlapping region is the area bounded by the two lines and lies below both of them.

3. Determine the vertices: The vertices of the feasible region are (0, 0) and the intersection of the two lines. To find the intersection, solve the system of equations:

   • y = -x + 5
   • y = 2x

   Substitute the second equation into the first:

   • 2x = -x + 5
   • 3x = 5
   • x = 5/3

   Substitute x = 5/3 back into y = 2x:

   • y = 2(5/3) = 10/3

   So the intersection point is (5/3, 10/3). The vertices of the feasible region are (0, 0) and (5/3, 10/3).

Common Mistakes and How to Avoid Them

• Forgetting to Reverse the Inequality Sign: When multiplying or dividing both sides of an inequality by a negative number, remember to reverse the inequality sign. This is a very common mistake.
• Using a Solid Line When it Should Be Dashed (or Vice Versa): Double-check whether the inequality includes ≤ or ≥ (solid line) or < or > (dashed line).
• Choosing a Test Point on the Boundary Line: Always choose a test point that is not on the boundary line. If you accidentally choose a point on the line, select a different test point.
• Shading the Wrong Half-Plane: Carefully substitute the test point into the inequality and determine whether the inequality is true or false. Shade the correct half-plane accordingly.
• Incorrectly Graphing the Boundary Line: Make sure you accurately graph the boundary line. Use the slope-intercept form to help you plot the line correctly. Double-check your y-intercept and slope.
• Not Understanding the Meaning of the Solution: Remember that the solution to a linear inequality is a region of points, not just a single point. The shaded region represents all the points (x, y) that satisfy the inequality.
• Errors in Arithmetic: Simple arithmetic errors can lead to incorrect results. Double-check your calculations, especially when solving for y in slope-intercept form or when substituting the test point.

Conclusion

Linear inequalities in two variables provide a powerful tool for modeling and solving real-world problems involving constraints and optimization. By understanding the key concepts, mastering the steps to solve and graph inequalities, and avoiding common mistakes, you can confidently tackle a wide range of applications. Whether you're optimizing production, planning a diet, or designing a structure, the principles of linear inequalities will prove invaluable. Remember to practice consistently and to visualize the solutions graphically to deepen your understanding. The journey of mastering linear inequalities is a rewarding one, unlocking a deeper appreciation for the beauty and utility of mathematics.