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Ch. 5 - Systems and Matrices
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 5 - Systems and MatricesProblem 39
Chapter 6, Problem 39

Graph the solution set of each system of inequalities.
2x + y > 2
x - 3y < 6

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Step 1: Rewrite each inequality in slope-intercept form (y = mx + b) to make graphing easier. For the first inequality, start with \(4x + y > 8\). Subtract \$4x\( from both sides to isolate \)y\(: \)y > -4x + 8$.
Step 2: For the second inequality, \(6x - y < 2\), subtract \$6x\( from both sides: \)-y < -6x + 2\(. Then multiply both sides by \)-1\( to solve for \)y\(, remembering to reverse the inequality sign: \)y > 6x - 2$.
Step 3: Graph the boundary lines for each inequality. For \(y > -4x + 8\), graph the line \(y = -4x + 8\). For \(y > 6x - 2\), graph the line \(y = 6x - 2\). Use dashed lines because the inequalities are strict (greater than, not greater than or equal to).
Step 4: Determine the solution region for each inequality by testing a point not on the line (commonly the origin (0,0) if it is not on the line). For \(y > -4x + 8\), check if (0,0) satisfies the inequality: \(0 > -4(0) + 8\) which is \(0 > 8\) (false), so shade the region opposite to where (0,0) lies. For \(y > 6x - 2\), check (0,0): \(0 > 6(0) - 2\) which is \(0 > -2\) (true), so shade the side containing (0,0).
Step 5: The solution set to the system is the intersection of the shaded regions from both inequalities. Identify and shade the overlapping region on the graph, which represents all points satisfying both inequalities simultaneously.

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Key Concepts

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

Graphing Linear Inequalities

Graphing linear inequalities involves plotting the boundary line of the corresponding linear equation and then shading the region that satisfies the inequality. For 'greater than' or 'less than' inequalities, the boundary line is dashed, indicating points on the line are not included. The shaded area represents all solutions that make the inequality true.
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System of Inequalities

A system of inequalities consists of two or more inequalities that are considered simultaneously. The solution set is the intersection of the individual solution regions, meaning only points that satisfy all inequalities are included. Graphing helps visualize this common solution area.
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Testing Points to Determine Shading

To decide which side of the boundary line to shade, select a test point not on the line (often the origin if not on the line) and substitute it into the inequality. If the inequality holds true, shade the side containing the test point; otherwise, shade the opposite side. This method ensures correct representation of the solution set.
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Related Practice
Textbook Question

Determine the system of equations illustrated in each graph. Write equations in standard form.

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Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components.

3x2 - y2 = 11

xy = 12

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Evaluate each determinant.

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Find each sum or difference, if possible. See Examples 2 and 3.

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Textbook Question

Graph the solution set of each system of inequalities.

x + y ≥ 0

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Textbook Question

Find the partial fraction decomposition for each rational expression. See Examples 1–4. (4x2 - 3x - 4)/(x3 + x2 - 2x)