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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 58
Chapter 6, Problem 58

Graph the solution set of each system of inequalities.
2y+x≥−52y+x\(\ge\)-5
y≤3+xy\(\le\)3+xx≤0x\(\le\)0
y≤0y\(\le\)0

Verified step by step guidance
1
Step 1: Start by graphing the inequality 2y + x \(\ge\) -5. Rewrite it in slope-intercept form (y = mx + b) by isolating y. Subtract x from both sides to get 2y \(\ge\) -x - 5, then divide every term by 2 to get y \(\ge\) -\(\frac{1}{2}\)x - \(\frac{5}{2}\). This represents a line with a slope of -\(\frac{1}{2}\) and a y-intercept of -\(\frac{5}{2}\). Shade the region above this line, as the inequality is \(\ge\).
Step 2: Next, graph the inequality y \(\le\) 3 + x. Again, rewrite it in slope-intercept form if necessary. This is already in the form y \(\le\) x + 3, which represents a line with a slope of 1 and a y-intercept of 3. Shade the region below this line, as the inequality is \(\le\).
Step 3: Graph the inequality x \(\le\) 0. This is a vertical line at x = 0. Shade the region to the left of this line, as the inequality is \(\le\).
Step 4: Graph the inequality y \(\le\) 0. This is a horizontal line at y = 0. Shade the region below this line, as the inequality is \(\le\).
Step 5: The solution set of the system of inequalities is the region where all the shaded areas overlap. Identify this region on the graph, which will be bounded by the lines you have drawn and shaded according to the inequalities.

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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 '≥' (greater than or equal to) and '≤' (less than or equal to) to indicate the range of possible solutions. Understanding how to interpret and manipulate inequalities is crucial for solving systems of inequalities.
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Graphing Linear Equations

Graphing linear equations involves plotting points on a coordinate plane that satisfy the equation. Each equation can be represented as a line, and the slope-intercept form (y = mx + b) is commonly used. Knowing how to graph these lines helps visualize the solution sets of inequalities, as the area above or below the line represents the solutions.
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Categorizing Linear Equations

Feasible Region

The feasible region is the area on a graph where all the inequalities in a system are satisfied simultaneously. It is typically bounded by the lines representing the inequalities and can be identified by shading the appropriate regions. Understanding how to find and interpret the feasible region is essential for solving systems of inequalities graphically.
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Related Practice
Textbook Question

Graph the solution set of each system of inequalities.

2x+3y≤122x+3y\(\le\)12

2x+3y>−62x+3y>-6

3x+y<43x+y\(\lt{4}\)

x≥0x\(\ge\)0

y≥0y\(\ge\)0

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Textbook Question

Use the determinant theorems to evaluate each determinant.

∣−102354−37829−544−110∣\(\left\)| \(\begin{matrix}\) -1 & 0 & 2 & 3 \\ 5 & 4 & -3 & 7 \\ 8 & 2 & 9 & -5 \\ 4 & 4 & -1 & 10 \(\end{matrix}\) \(\right\)|

Textbook Question

Use the determinant theorems to evaluate each determinant.

∣4002−103024010012∣\(\left\)| \(\begin{matrix}\) 4 & 0 & 0 & 2 \\ -1 & 0 & 3 & 0 \\ 2 & 4 & 0 & 1 \\ 0 & 0 & 1 & 2 \(\end{matrix}\) \(\right\)|

Textbook Question

Graph the solution set of each system of inequalities.

x≤4x\(\le\)4

x≥0x\(\ge\)0

y≥0y\(\ge\)0

x+2y≥2x+2y\(\ge\)2

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Textbook Question

Find each product, if possible.

[22−183270][8−1091202]\(\left\)[ \(\begin{matrix}\) \(\sqrt{2}\) & \(\sqrt{2}\) & -\(\sqrt{18}\) \\ \(\sqrt{3}\) & \(\sqrt{27}\) & 0 \(\end{matrix}\) \(\right\)] \(\left\)[ \(\begin{matrix}\) 8 & -10 \\ 9 & 12 \\ 0 & 2 \(\end{matrix}\) \(\right\)]

Textbook Question

Find each product, if possible.

[3−41502][−142]\(\left\)[ \(\begin{matrix}\) 3 & -4 & 1 \\ 5 & 0 & 2 \(\end{matrix}\) \(\right\)] \(\left\)[ \(\begin{matrix}\) -1 \\ 4 \\ 2 \(\end{matrix}\) \(\right\)]