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$X-2y\geq0$ $x+2y\leq4$ In which of the following does the shaded region represent the solution set in the $xy$ -plane to the system of inequalities?

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Rational functions: asymptotes and excluded values

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Q.

X-2y\geq0

x+2y\leq4

In which of the following does the shaded region represent the solution set in the

xy

-plane to the system of inequalities?

Identify and Rewrite Inequality: Identify the first inequality and rewrite it in slope-intercept form $y = mx + b$ to graph it. The first inequality is $X - 2y \geq 0$ . To get $y$ by itself, add $2y$ to both sides and then divide by $-2$ . The inequality becomes $y \leq \frac{X}{2}$ . This represents a line with a slope of $\frac{1}{2}$ and $y$ -intercept at $0$ . The inequality sign indicates that the region above the line is not included, so we will shade below the line.
Graph First Inequality: Graph the first inequality. The line $y = \frac{X}{2}$ will be a dashed line because the inequality is not strict (it includes the line itself). The region below this line (including the line) will be shaded to represent the solution set for the first inequality.
Identify and Rewrite Second Inequality: Identify the second inequality and rewrite it in slope-intercept form. The second inequality is $x + 2y \leq 4$ . To get $y$ by itself, subtract $x$ from both sides and then divide by $2$ . The inequality becomes $y \leq -\frac{X}{2} + 2$ . This represents a line with a slope of $-\frac{1}{2}$ and $y$ -intercept at $2$ . The inequality sign indicates that the region below the line is included, so we will shade below the line.
Graph Second Inequality: Graph the second inequality. The line $y = -\frac{X}{2} + 2$ will be a solid line because the inequality includes the line itself. The region below this line will be shaded to represent the solution set for the second inequality.
Determine Intersection of Shaded Regions: Determine the intersection of the shaded regions. The solution to the system of inequalities will be the region where the shading from the first inequality (below $y = \frac{X}{2}$ ) and the shading from the second inequality (below $y = -\frac{X}{2} + 2$ ) overlap. This is the region that satisfies both inequalities simultaneously.
Final Shaded Region: The final shaded region is the area that is below both lines on the graph. This is the region that satisfies both $X - 2y \geq 0$ and $x + 2y \leq 4$ . The region is bounded by the lines $y = \frac{X}{2}$ and $y = -\frac{X}{2} + 2$ and includes the area where these two inequalities overlap.

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