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 31

Textbook Question

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

Verified step by step guidance

Graph the line y = 4x - 2. This line has a slope of 4 and a y-intercept of -2. Since the inequality is y > 4x - 2, shade the region above this line.

Graph the line y = -3x + 9. This line has a slope of -3 and a y-intercept of 9. Since the inequality is y < -3x + 9, shade the region below this line.

Identify the region where the shaded areas from both inequalities overlap. This overlapping region is the solution set for the system of inequalities.

Check if the lines are parallel or intersect. If they intersect, find the point of intersection by solving the equations 4x - 2 = -3x + 9.

Verify that the point of intersection, if any, is not included in the solution set since the inequalities are strict (using > and <).

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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. In this context, they indicate regions on a graph where one variable is greater than or less than another. Understanding how to interpret and graph inequalities is crucial for visualizing solution sets.

Graphing Linear Inequalities

Graphing linear inequalities involves plotting the boundary line of the inequality and determining which side of the line represents the solution set. A dashed line indicates that points on the line are not included in the solution, while a solid line means they are. The shading of the graph shows where the inequality holds true.

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. Analyzing systems of inequalities helps in finding common solutions and understanding the relationships between different constraints.