Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

7. Systems of Equations & Matrices

Graphing Systems of Inequalities

College Algebra

7. Systems of Equations & Matrices

Graphing Systems of Inequalities

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1:26 minutes

Problem 27

Textbook Question

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. 3x+6y≤6, 2x+y≤8

Verified step by step guidance

Step 1: Start by graphing the inequality 2x + 4y \leq 8. First, rewrite it in slope-intercept form (y = mx + b) by solving for y.

Step 2: Divide the entire inequality 2x + 4y \leq 8 by 4 to simplify it to y \leq -\frac{1}{2}x + 2.

Step 3: Graph the line y = -\frac{1}{2}x + 2. Use a solid line because the inequality is \leq, indicating that points on the line are included in the solution set.

Step 4: Shade the region below the line y = -\frac{1}{2}x + 2, as the inequality is y \leq -\frac{1}{2}x + 2.

Step 5: Repeat the process for the second inequality 4x + y \leq 4. Solve for y to get y \leq -4x + 4, graph the line, and shade the region below it. The solution set is the intersection of the shaded regions.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Inequalities

Inequalities are mathematical expressions that show the relationship between two values when they are not equal. They use symbols such as ≤ (less than or equal to) and ≥ (greater than or equal to) to indicate the range of possible solutions. Understanding how to manipulate and graph inequalities is crucial for solving systems of inequalities.

Graphing Systems of Inequalities

Graphing systems of inequalities involves plotting each inequality on a coordinate plane to visualize the solution set. The area where the shaded regions of the inequalities overlap represents the solutions that satisfy all inequalities in the system. This graphical representation helps in identifying feasible solutions and understanding the relationships between the inequalities.

Feasible Region

The feasible region is the area on a graph where all the constraints of a system of inequalities are satisfied. It is typically bounded by the lines representing the inequalities and can be unbounded in some cases. Identifying the feasible region is essential for determining the solutions to the system and for optimization problems in algebra.