Question

graph the following system of inequalities on the coordinate plane and shade the resulting solution region: \\(\

$$\begin{cases} y > 2x - 3 \\\\ y \\leq -x + 4 \\end{cases}$$

\\)

Explanation:

Step1: Graph \( y > 2x - 3 \)

First, graph the line \( y = 2x - 3 \). This is a straight line with slope \( 2 \) and y - intercept \( - 3 \). Since the inequality is \( y>2x - 3 \), we use a dashed line (because the inequality is strict, \( y\) is not equal to \( 2x - 3 \)) to represent the boundary. Then, to determine which side of the line to shade, we can test a point not on the line. Let's use the origin \( (0,0) \). Substitute \( x = 0 \) and \( y = 0 \) into the inequality: \( 0>2(0)-3\), which simplifies to \( 0 > - 3 \), a true statement. So we shade the region above the dashed line \( y = 2x - 3 \).

Step2: Graph \( y\leq -x + 4 \)

Next, graph the line \( y=-x + 4 \). This line has a slope of \( - 1 \) and a y - intercept of \( 4 \). Since the inequality is \( y\leq -x + 4 \), we use a solid line (because the inequality is non - strict, \( y\) can be equal to \( -x + 4 \)) to represent the boundary. Again, test a point not on the line, such as the origin \( (0,0) \). Substitute \( x = 0 \) and \( y = 0 \) into the inequality: \( 0\leq-(0)+4\), which simplifies to \( 0\leq4 \), a true statement. So we shade the region below (or on) the solid line \( y=-x + 4 \).

Step3: Find the solution region

The solution region of the system of inequalities is the region that is shaded for both inequalities simultaneously. That is, the region that is above the dashed line \( y = 2x - 3 \) and below (or on) the solid line \( y=-x + 4 \). To find the intersection point of the two lines \( y = 2x-3 \) and \( y=-x + 4 \), we set them equal to each other:

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Substitute \( x=\frac{7}{3} \) into \( y=-x + 4 \), we get \( y=-\frac{7}{3}+4=\frac{-7 + 12}{3}=\frac{5}{3} \). So the two lines intersect at \( (\frac{7}{3},\frac{5}{3}) \). The solution region is the area bounded by the two lines (with the appropriate line types) and including the part that satisfies both inequalities.

Answer:

To graph the system:

1. For \( y>2x - 3 \): Draw a dashed line \( y = 2x-3 \), shade above it.
2. For \( y\leq -x + 4 \): Draw a solid line \( y=-x + 4 \), shade below (and on) it.
3. The solution region is the overlapping shaded area, bounded by the dashed line \( y = 2x - 3 \) (above it), the solid line \( y=-x + 4 \) (below/on it), and their intersection point \( (\frac{7}{3},\frac{5}{3}) \).