Graph the solution set of each system of inequalities or indicate that the system has no solution. {x≥0y≥02x+5y\u003c103x+4y≤12\begin{cases} x \geq 0 \\ y \geq 0 \\ 2x + 5y \u003c 10 \\ 3x + 4y \leq 12 \end{cases} ⎩⎨⎧x≥0y≥02x+5y\u003c103x+4y≤12

Identify the inequalities in the system: $$x \geq 1$$, $$y \geq -1$$, $$x + 6y \u003c 15$$, and $$2x + y \leq 5. $$Graph the boundary lines for each inequality by converting inequalities to equations: $$x = 1$$, $$y = -1$$, $$x + 6y = 15$$, and $$2x + y = 5. $$Determine the shading direction for each inequality: For $$x \geq 1$$, shade to the right of the vertical line $$x=1$$; for $$y \geq -1$$, shade above the horizontal line $$y=-1$$; for $$x + 6y \u003c 15$$, shade below the line $$x + 6y = 15$$; for $$2x + y \leq 5$$, shade below or on the line $$2x + y = 5. $$Find the intersection region where all shaded areas overlap. This region represents the solution set to the system of inequalities. Check if the intersection region is non-empty. If it exists, the solution set is the overlapping shaded area; if not, the system has no solution.

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 59

6장, 문제 59

Graph the solution set of each system of inequalities or indicate that the system has no solution.
$\begin{cases}\) x $\geq$ 0 \\ y $\geq$ 0 \\ 2x + 5y < 10 \\ 3x + 4y $\leq$ 12 \(\end{cases}$

검증된 단계별 안내

Identify the inequalities in the system: $x \geq 1$, $y \geq -1$, $x + 6y < 15$, and $2x + y \leq 5$.

Graph the boundary lines for each inequality by converting inequalities to equations: $x = 1$, $y = -1$, $x + 6y = 15$, and $2x + y = 5$.

Determine the shading direction for each inequality: For $x \geq 1$, shade to the right of the vertical line $x=1$; for $y \geq -1$, shade above the horizontal line $y=-1$; for $x + 6y < 15$, shade below the line $x + 6y = 15$; for $2x + y \leq 5$, shade below or on the line $2x + y = 5$.

Find the intersection region where all shaded areas overlap. This region represents the solution set to the system of inequalities.

Check if the intersection region is non-empty. If it exists, the solution set is the overlapping shaded area; if not, the system has no solution.

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 Linear Inequalities

Graphing linear inequalities involves plotting the boundary line of the inequality and shading the region that satisfies the inequality. For strict inequalities (< or >), the boundary is dashed, while for inclusive inequalities (≤ or ≥), the boundary is solid. This visual representation helps identify all possible solutions.

System of Inequalities

A system of inequalities consists of multiple inequalities that must be satisfied simultaneously. The solution set is the intersection of the individual solution regions of each inequality. Understanding how to find this common region is essential for solving such systems.

Boundary Conditions and Feasible Region

Boundary conditions like x ≥ 1 and y ≥ -1 restrict the solution to specific quadrants or areas on the coordinate plane. The feasible region is the overlapping area that meets all inequalities, representing all possible solutions. Identifying this region is key to solving and interpreting the system.

Graph the solution set of each system of inequalities or indicate that the system has no solution. {x≥0y≥02x+5y<103x+4y≤12\(\begin{cases}\) x \(\geq\) 0 \\ y \(\geq\) 0 \\ 2x + 5y < 10 \\ 3x + 4y \(\leq\) 12 \(\end{cases}\) ⎩⎨⎧x≥0y≥02x+5y<103x+4y≤12

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Graphing Linear Inequalities

System of Inequalities

Boundary Conditions and Feasible Region

Graph the solution set of each system of inequalities or indicate that the system has no solution.
$\begin{cases}\) x \(\geq\) 0 \\ y \(\geq\) 0 \\ 2x + 5y < 10 \\ 3x + 4y \(\leq\) 12 \(\end{cases}$