Graph the solution set of each system of inequalities.y≥3xy\$$ge3^{x}y$$≥3xy≥2y\ge2y≥2

Ch. 5 - Systems and Matrices

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Lial 13th Edition

Ch. 5 - Systems and Matrices

문제 64

6장, 문제 64

Graph the solution set of each system of inequalities.
$y$\ge$3^{x}$
$y$\ge$2$

검증된 단계별 안내

Step 1: Identify the inequalities in the system. The system consists of two inequalities: $ y \geq 3^x $ and $ y \geq 2 $.

Step 2: Graph the first inequality $ y \geq 3^x $. Start by graphing the boundary line $ y = 3^x $, which is an exponential function. Plot points for several values of $ x $ to get the shape of the curve, such as $ (0, 1), (1, 3), (2, 9) $, and so on. Since the inequality is $ \geq $, shade the region above the curve.

Step 3: Graph the second inequality $ y \geq 2 $. This is a horizontal line at $ y = 2 $. Since the inequality is $ \geq $, shade the region above this line.

Step 4: Determine the solution set by finding the intersection of the shaded regions from both inequalities. The solution set is where the shaded region from $ y \geq 3^x $ overlaps with the shaded region from $ y \geq 2 $.

Step 5: Clearly mark the solution set on the graph. This is the region that satisfies both inequalities, and it will be the area above both the curve $ y = 3^x $ and the line $ y = 2 $.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

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주요 개념

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

Inequalities

Inequalities are mathematical expressions that show the relationship between two values when they are not equal. They use symbols such as '≥' (greater than or equal to) and '≤' (less than or equal to) to indicate the range of possible solutions. Understanding how to interpret and graph inequalities is crucial for visualizing solution sets in coordinate systems.

Graphing Exponential Functions

Exponential functions, such as y = 3^x, represent growth or decay processes and are characterized by a constant base raised to a variable exponent. When graphing these functions, it is important to recognize their shape, which typically includes a horizontal asymptote and increases rapidly for positive x-values. This understanding helps in determining the region of the graph that satisfies the inequality.

Shading Regions in Graphs

When graphing inequalities, shading is used to represent the solution set. For a given inequality, the area above (or below) the line is shaded to indicate all the points that satisfy the inequality. Knowing how to correctly shade the regions based on the inequality symbols is essential for accurately depicting the solution set on a graph.

Graph the solution set of each system of inequalities.y≥3xy\(\ge\)3^{x}y≥3xy≥2y\(\ge\)2y≥2

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Inequalities

Graphing Exponential Functions

Shading Regions in Graphs

Graph the solution set of each system of inequalities.
$y\(\ge\)3^{x}$
$y\(\ge\)2$