Graph the solution set of each system of inequalities.2x+3y≤122x+3y\le122x+3y\u003e−62x+3y\u003e-63x+y\u003c43x+y\lt{4}x≥0x\ge0y≥0y\ge0

Ch. 5 - Systems and Matrices

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

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Lial 13th Edition

Ch. 5 - Systems and Matrices

Problem 59

Chapter 6, Problem 59

Graph the solution set of each system of inequalities.
$2 x plus 3 y is less than or equal to 12$
$2 x + 3 y > - 6$
$3 x plus y is less than 4$
$x is greater than or equal to 0$
$y is greater than or equal to 0$

Verified step by step guidance

Rewrite each inequality in slope-intercept form (y = mx + b) to better understand the boundary lines. For example, for the inequality \[2x + 3y \le 12\], solve for \[y\]: \[3y \le 12 - 2x\], then \[y \le \frac{12 - 2x}{3}\].

Graph the boundary lines for each inequality. Use a solid line for inequalities with $ \le $ or $ \ge $ because the boundary is included, and a dashed line for strict inequalities $ < $ or $ > $ because the boundary is not included.

Determine which side of each boundary line to shade by choosing a test point (commonly the origin $ (0,0) $ if it is not on the line) and checking if it satisfies the inequality. Shade the region that satisfies the inequality.

For the inequalities \[x \ge 0\] and \[y \ge 0\], shade the regions to the right of the y-axis and above the x-axis respectively, since these represent the first quadrant constraints.

Find the intersection of all shaded regions from the inequalities. The solution set to the system is the area where all these shaded regions overlap. This common region represents all points that satisfy every inequality 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 first graphing the related linear equation as a boundary line. The inequality symbol determines whether the boundary is solid (≤ or ≥) or dashed (< or >). Then, shading the region that satisfies the inequality shows the solution set. This visual approach helps identify all points that make the inequality true.

Systems of Inequalities

A system of inequalities consists of multiple inequalities that must be satisfied simultaneously. The solution set is the intersection of the individual solution regions. Graphing each inequality and finding the overlapping shaded area reveals all points that satisfy the entire system.

Coordinate Plane and Quadrants

The coordinate plane is divided into four quadrants by the x- and y-axes. Inequalities like x ≥ 0 and y ≥ 0 restrict solutions to specific quadrants, often the first quadrant where both x and y are nonnegative. Understanding this helps limit the graphing region and interpret the solution set correctly.

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Textbook Question

Solve each problem using a system of equations in two variables. See Example 6. Find two numbers whose ratio is 9 to 2 and whose product is 162.

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Textbook Question

Use the determinant theorems to evaluate each determinant.

$∣ \begin{matrix} 4 & 0 & 0 & 2 \\ - 1 & 0 & 3 & 0 \\ 2 & 4 & 0 & 1 \\ 0 & 0 & 1 & 2 \end{matrix} ∣$

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Textbook Question

Solve each problem using a system of equations in two variables. See Example 6. Find two numbers whose ratio is 4 to 3 and are such that the sum of their squares is 100.

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Textbook Question

Find each product, if possible.

$[\begin{matrix} \sqrt{2} & \sqrt{2} & - \sqrt{18} \\ \sqrt{3} & \sqrt{27} & 0 \end{matrix}] [\begin{matrix} 8 & - 10 \\ 9 & 12 \\ 0 & 2 \end{matrix}]$