Graph the solution set of each system of inequalities or indicate that the system has no solution. {x2+y2≤16x+y\u003e2\begin{cases} $$x^2$$ + $$y^2$$ \leq 16 \\ x + y \u003e 2 \end{cases} {x2+y2≤16x+y\u003e2

Ch. 5 - Systems of Equations and Inequalities

Blitzer8th EditionCollege AlgebraISBN: 9780136970514당신이 사용하는 게 아니라요?교과서 변경

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Blitzer 8th Edition

Ch. 5 - Systems of Equations and Inequalities

문제 47

6장, 문제 47

Graph the solution set of each system of inequalities or indicate that the system has no solution.
$\begin{cases}\) x^2 + y^2 $\leq$ 16 \\ x + y > 2 \(\end{cases}$

검증된 단계별 안내

Identify the first inequality: $x^{2} + y^{2} \leq 36$. This represents all points inside or on the circle centered at the origin $(0,0)$ with radius $6$, since $\sqrt{36} = 6$.

Identify the second inequality: $3x + y > 6$. This is a linear inequality representing all points above the line $3x + y = 6$.

Rewrite the linear inequality in slope-intercept form to better understand the boundary line: $y > -3x + 6$. This line has a slope of $-3$ and a y-intercept at $(0,6)$.

Graph the circle $x^{2} + y^{2} = 36$ as a solid boundary because the inequality includes equality ($\leq$), meaning points on the circle are included.

Graph the line $y = -3x + 6$ as a dashed line because the inequality is strict ($>$), meaning points on the line are not included. Then shade the region above this line where $y > -3x + 6$.

The solution set to the system is the intersection of the shaded region inside the circle and above the line. This means you look for points that satisfy both inequalities simultaneously.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

주요 개념

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

Graphing Circles and Inequalities

The inequality x² + y² ≤ 36 represents all points inside or on the circle centered at the origin with radius 6. Understanding how to graph this region involves recognizing the circle's boundary (x² + y² = 36) and shading the interior where the inequality holds true.

Graphing Linear Inequalities

The inequality 3x + y > 6 represents a half-plane above the line 3x + y = 6. To graph it, first plot the boundary line, then determine which side satisfies the inequality by testing a point, shading the region where the inequality is true.

Solution Set of a System of Inequalities

The solution set of the system is the intersection of the regions satisfying each inequality. It includes points inside the circle and above the line simultaneously. Identifying this overlap is key to correctly graphing the system and understanding if solutions exist.

Graph the solution set of each system of inequalities or indicate that the system has no solution. {x2+y2≤16x+y>2\(\begin{cases}\) x^2 + y^2 \(\leq\) 16 \\ x + y > 2 \(\end{cases}\) {x2+y2≤16x+y>2

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Graphing Circles and Inequalities

Graphing Linear Inequalities

Solution Set of a System of Inequalities

Graph the solution set of each system of inequalities or indicate that the system has no solution.
$\begin{cases}\) x^2 + y^2 \(\leq\) 16 \\ x + y > 2 \(\end{cases}$