Graph the solution set of each system of inequalities.
4x - 3y ≤ 12
y ≤ x²

검증된 단계별 안내

Step 1: Identify each inequality in the system. The first inequality is \(4y - 2x \leq 15\) and the second inequality is \(y \geq -x^{2} + 2\).

Step 2: Rewrite the first inequality in slope-intercept form (\(y = mx + b\)) to make graphing easier. Start by isolating \(y\): \(4y \leq 2x + 15\), then divide both sides by 4 to get \(y \leq \frac{1}{2}x + \frac{15}{4}\).

Step 3: Graph the line \(y = \frac{1}{2}x + \frac{15}{4}\). Since the inequality is \(\leq\), shade the region below or on this line.

Step 4: Graph the parabola \(y = -x^{2} + 2\). Since the inequality is \(y \geq -x^{2} + 2\), shade the region above or on this parabola.

Step 5: The solution set to the system is the region where the shaded areas from both inequalities overlap. This overlapping region satisfies both inequalities simultaneously.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

12m

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주요 개념

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

Graphing Linear Inequalities

A linear inequality like 4y - 2x ≤ 15 represents a half-plane on the coordinate plane. To graph it, first rewrite it in slope-intercept form (y ≤ (1/2)x + 15/4), then draw the boundary line (solid for ≤ or ≥) and shade the region that satisfies the inequality.

Graphing Quadratic Inequalities

A quadratic inequality such as y ≥ -x² + 2 involves a parabola. The graph of y = -x² + 2 is a downward-opening parabola shifted up by 2 units. The inequality y ≥ -x² + 2 means shading the region on or above this parabola.

Solution Set of a System of Inequalities

The solution set of a system of inequalities is the intersection of the regions satisfying each inequality. Graph each inequality separately, then identify the overlapping shaded area that meets all conditions simultaneously.

Graph the solution set of each system of inequalities. 4x - 3y ≤ 12 y ≤ x2