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 \( y = 2x - 3 \). The slope \( m = 2 \) and the y - intercept \( b=-3 \). Since the inequality is \( y>2x - 3 \), the line should be dashed (because the inequality is strict, \( y\) is not equal to \( 2x - 3 \)). To determine which side to shade, we can use a test point, for example, the origin \((0,0)\). Substitute \( x = 0,y = 0 \) 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 \( y=-x + 4 \). The slope \( m=-1 \) and the y - intercept \( b = 4 \). Since the inequality is \( y\leq -x + 4 \), the line should be solid (because \( y\) can be equal to \( -x + 4 \)). Use the origin \((0,0)\) as a test point. Substitute \( x = 0,y = 0 \) 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 intersection region

The solution to the system of inequalities is the region that is shaded by both inequalities. So we look for the area 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 \( y=-x + 4 \), set \( 2x-3=-x + 4 \). Add \( x \) to both sides: \( 3x-3 = 4 \). Add 3 to both sides: \( 3x=7 \), so \( x=\frac{7}{3}\approx2.33 \). Substitute \( x = \frac{7}{3}\) into \( y=-x + 4 \), we get \( y=-\frac{7}{3}+4=\frac{-7 + 12}{3}=\frac{5}{3}\approx1.67 \). So the two lines intersect at the point \((\frac{7}{3},\frac{5}{3})\). The solution region is the area that is above the dashed line \( y = 2x-3 \), below (or on) the solid line \( y=-x + 4 \), and bounded by their intersection point.

Answer:

To graph the system:

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