Graph each inequality. x \u003c 3 + 2y

Ch. 5 - Systems and Matrices

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Lial 13th Edition

Ch. 5 - Systems and Matrices

문제 16

6장, 문제 16

Graph each inequality. x < 3 + 2y

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Rewrite the inequality to isolate \( y \) on one side. Starting with \( x < 3 + 2y \), subtract 3 from both sides to get \( x - 3 < 2y \).

Divide both sides of the inequality by 2 to solve for \( y \): \( \frac{x - 3}{2} < y \). This can be rewritten as \( y > \frac{x - 3}{2} \).

Identify the boundary line for the inequality, which is \( y = \frac{x - 3}{2} \). Since the inequality is strict (\( > \)), the boundary line will be dashed on the graph.

Determine which side of the boundary line to shade. Pick a test point not on the line, such as \( (0,0) \), and substitute into the inequality \( y > \frac{x - 3}{2} \). If the test point satisfies the inequality, shade that side; otherwise, shade the opposite side.

Draw the dashed boundary line \( y = \frac{x - 3}{2} \) on the coordinate plane and shade the region above the line (if the test point satisfies the inequality) to represent all solutions to the inequality.

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주요 개념

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

Inequalities in Two Variables

An inequality involving two variables, like x and y, represents a region in the coordinate plane where the inequality holds true. Instead of a single line, the solution is a shaded area that satisfies the inequality condition.

Graphing Linear Inequalities

To graph a linear inequality, first rewrite it in a form that relates x and y, then graph the boundary line (using equality). Use a solid line if the inequality includes equality (≤ or ≥), or a dashed line if it does not (< or >).

Testing Points to Determine Shading

After drawing the boundary line, select a test point not on the line (commonly (0,0)) to check if it satisfies the inequality. Shade the side of the line where the inequality holds true, representing all solutions.