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

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Blitzer 8th Edition

Ch. 5 - Systems of Equations and Inequalities

Problema 59

Capitolo 6, Problema 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}$

Guida verificata passo dopo passo

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.

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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.

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Linear Inequalities

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.

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Systems of Inequalities

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.

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Categorizing Linear Equations

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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

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Concetti chiave

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}$