Question

y < -2x + 7\
y > 2x - 5

Explanation:

Step1: Analyze the first inequality \( y \leq -2x + 7 \)

The equation of the boundary line is \( y=-2x + 7 \). The slope \( m=-2 \) and the y - intercept \( b = 7 \). Since the inequality is \( y\leq-2x + 7 \), we draw a solid line (because the inequality includes equality) and shade the region below the line.

Step2: Analyze the second inequality \( y>2x - 5 \)

The equation of the boundary line is \( y = 2x-5 \). The slope \( m = 2 \) and the y - intercept \( b=-5 \). Since the inequality is \( y>2x - 5 \), we draw a dashed line (because the inequality does not include equality) and shade the region above the line.

Step3: Find the intersection of the two regions

To find the solution of the system of inequalities, we need to find the region that is shaded for both inequalities. First, let's find the intersection point of the two boundary lines \( y=-2x + 7 \) and \( y=2x-5 \). Set \( -2x + 7=2x-5 \).
Add \( 2x \) to both sides: \( 7 = 4x-5 \)
Add 5 to both sides: \( 12 = 4x \)
Divide both sides by 4: \( x = 3 \)
Substitute \( x = 3 \) into \( y=2x-5 \), we get \( y=2\times3-5=1 \). So the intersection point of the two lines is \( (3,1) \).

The solution region is the set of all points that are below (or on) the line \( y=-2x + 7 \) and above the line \( y = 2x-5 \).

Answer:

The solution to the system of inequalities \(

$$\begin{cases}y\leq - 2x + 7\\y>2x-5\end{cases}$$

\) is the region that is below (or on) the line \( y=-2x + 7 \) and above the line \( y = 2x-5 \), with the intersection point of the two lines being \( (3,1) \). The boundary line \( y=-2x + 7 \) is solid and \( y = 2x-5 \) is dashed.