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Graph the feasible region for the system of inequalities. \( y>4 x-3 \) \( y<-2 x+5 \)

Ask by Adkins Davison. in the United States
Feb 26,2025

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Answer

To graph the feasible region for the inequalities \( y>4x-3 \) and \( y<-2x+5 \), first find the intersection point of the lines \( y=4x-3 \) and \( y=-2x+5 \). Solve the system: \( 4x - 3 = -2x + 5 \) \( 6x = 8 \) \( x = \frac{4}{3} \) Substitute \( x = \frac{4}{3} \) into \( y = 4x - 3 \): \( y = 4(\frac{4}{3}) - 3 = \frac{16}{3} - 3 = \frac{7}{3} \) So, the intersection point is \( (\frac{4}{3}, \frac{7}{3}) \). Next, determine the regions where both inequalities are satisfied. The feasible region is the area between the two lines where \( y > 4x - 3 \) and \( y < -2x + 5 \) overlap. Graph the lines \( y = 4x - 3 \) and \( y = -2x + 5 \) on a coordinate plane. Shade the area above \( y = 4x - 3 \) and below \( y = -2x + 5 \) to represent the feasible region. The feasible region is the area bounded by these two lines and includes all points that satisfy both inequalities simultaneously.

Solution

To graph the feasible region for the system of inequalities \( y>4x-3 \) and \( y<-2x+5 \), we need to find the intersection points of the lines represented by the inequalities and then determine the regions where both inequalities are satisfied. 1. First, let's find the intersection points of the lines represented by the inequalities: - For the inequality \( y>4x-3 \), we can rewrite it as \( y-4x+3>0 \). - For the inequality \( y<-2x+5 \), we can rewrite it as \( y+2x-5<0 \). 2. Next, we need to find the intersection points of the lines represented by the inequalities. To do this, we can set the two inequalities equal to each other and solve for \( x \) and \( y \). 3. Once we have the intersection points, we can determine the feasible region by identifying the areas where both inequalities are satisfied. Let's proceed with finding the intersection points and determining the feasible region.

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To graph the feasible region for this system of inequalities, first, you need to graph the lines \( y = 4x - 3 \) and \( y = -2x + 5 \). The line \( y = 4x - 3 \) has a slope of 4, indicating it will rise steeply. The y-intercept is at -3, so it crosses the y-axis there. Draw this line as a dashed line since the inequality is strict (greater than). Next, graph the line \( y = -2x + 5 \), which has a slope of -2, making it decline as you move to the right. This line will intersect the y-axis at 5. Again, use a dashed line for this inequality. Now, to find the feasible region, shade the area above the line \( y = 4x - 3 \) and below the line \( y = -2x + 5 \). The intersection points of these lines can be calculated to find the corners of the feasible region if required. This shaded region is where both inequalities are satisfied!

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