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

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Lial 13th Edition

Ch. 5 - Systems and Matrices

문제 59

6장, 문제 59

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

검증된 단계별 안내

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.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

12m

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

Graph the solution set of each system of inequalities.2x+3y≤122x+3y\(\le\)122x+3y>−62x+3y>-63x+y<43x+y\(\lt{4}\)x≥0x\(\ge\)0y≥0y\(\ge\)0