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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1:22 minutes

Problem 33

Textbook Question

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. x+2y≤4, y≥x−3

Verified step by step guidance

Step 1: Start by graphing the inequality x + 5y \leq 8. First, find the boundary line by setting x + 5y = 8.

Step 2: To graph the boundary line, find the x-intercept by setting y = 0, which gives x = 8. Then find the y-intercept by setting x = 0, which gives y = \frac{8}{5}.

Step 3: Draw the line through the intercepts (8, 0) and (0, \frac{8}{5}). Since the inequality is \leq, shade the region below the line.

Step 4: Next, graph the inequality y \geq x - 6. Start by graphing the boundary line y = x - 6.

Step 5: Draw the line through the intercepts (6, 0) and (0, -6). Since the inequality is \geq, shade the region above the line. The solution set is the overlapping shaded region.

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Key Concepts

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

Graphing Inequalities

Graphing inequalities involves representing the solutions of an inequality on a coordinate plane. Each inequality can be graphed as a line, with a solid line indicating 'less than or equal to' or 'greater than or equal to,' and a dashed line for 'less than' or 'greater than.' The area that satisfies the inequality is shaded, showing all possible solutions.

System of Inequalities

A system of inequalities consists of two or more inequalities that are considered simultaneously. The solution set is the region where the shaded areas of all inequalities overlap. Understanding how to find this intersection is crucial for determining the feasible solutions that satisfy all conditions of the system.

Feasible Region

The feasible region is the area on a graph where all the inequalities in a system are satisfied. It is bounded by the lines of the inequalities and can be unbounded in some cases. Identifying this region is essential for solving optimization problems, where one seeks to maximize or minimize a particular objective function within the constraints defined by the inequalities.