Graph the solution set of each system of inequalities.
x + y ≥ 0
2x - y ≥ 3

검증된 단계별 안내

Step 1: Understand the system of inequalities. You are given two inequalities: \(x - y \geq 0\) and \(4x + 5y \geq 9\). Your goal is to graph the solution set that satisfies both inequalities simultaneously.

Step 2: Rewrite each inequality in slope-intercept form (\(y = mx + b\)) to make graphing easier. For the first inequality, \(x - y \geq 0\), rearrange to isolate \(y\): \(-y \geq -x\) which simplifies to \(y \leq x\) after multiplying both sides by \(-1\) and reversing the inequality sign.

Step 3: For the second inequality, \(4x + 5y \geq 9\), isolate \(y\): \(5y \geq 9 - 4x\), then divide both sides by 5 to get \(y \geq \frac{9}{5} - \frac{4}{5}x\).

Step 4: Graph the boundary lines \(y = x\) and \(y = \frac{9}{5} - \frac{4}{5}x\). Use a solid line for each because the inequalities include equality (\(\geq\) or \(\leq\)).

Step 5: Determine the solution regions for each inequality by testing points on either side of the boundary lines. For \(y \leq x\), shade the region below or on the line \(y = x\). For \(y \geq \frac{9}{5} - \frac{4}{5}x\), shade the region above or on the line \(y = \frac{9}{5} - \frac{4}{5}x\). The solution set is the intersection of these shaded regions.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

11m

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주요 개념

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

Graphing Linear Inequalities

Graphing linear inequalities involves plotting the boundary line of the corresponding equation and then shading the region that satisfies the inequality. The boundary line is solid if the inequality includes equality (≥ or ≤) and dashed if it does not (> or <). This visual representation helps identify all solutions that satisfy the inequality.

System of Inequalities

A system of inequalities consists of two or more inequalities considered simultaneously. The solution set is the intersection of the regions that satisfy each inequality individually. Graphing the system helps find the common area where all inequalities hold true.

Testing Points to Determine Solution Regions

To decide which side of the boundary line to shade, select a test point not on the line (often the origin) 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 accurate graphing of the solution set.

Graph the solution set of each system of inequalities. x + y ≥ 0 2x - y ≥ 3