Question

1. \

$$\begin{cases} y > -2x + 5 \\\\ y \\leq x - 3 \\end{cases}$$

(and some handwritten notes and a coordinate grid below)

Explanation:

Step1: Analyze the inequalities

We have the system of inequalities: \( y > -2x + 5 \) and \( y \leq x - 3 \). First, we need to graph the boundary lines for each inequality.

Step2: Graph \( y = -2x + 5 \)

The slope of this line is \( -2 \) and the y-intercept is \( 5 \). Since the inequality is \( y > -2x + 5 \), we draw a dashed line (because the inequality is strict, \( > \)) and shade the region above the line.

Step3: Graph \( y = x - 3 \)

The slope of this line is \( 1 \) and the y-intercept is \( -3 \). Since the inequality is \( y \leq x - 3 \), we draw a solid line (because the inequality is non - strict, \( \leq \)) and shade the region below the line.

Step4: Find the intersection region

The solution to the system of inequalities is the region that is shaded for both inequalities. To find the intersection point of the two lines \( y=-2x + 5\) and \(y=x - 3\), we set them equal to each other:

$$
LATEXBLOCK0
$$

Substitute \(x = \frac{8}{3}\) into \(y=x - 3\), we get \(y=\frac{8}{3}-3=\frac{8 - 9}{3}=-\frac{1}{3}\approx - 0.33\)

Now, we can check the regions. For \(y > -2x+5\), when \(x = 0\), \(y>5\), so we shade above \(y=-2x + 5\). For \(y\leq x - 3\), when \(x = 0\), \(y\leq - 3\), so we shade below \(y=x - 3\). The overlapping region (the solution set) is the area that is above the dashed line \(y=-2x + 5\) and below the solid line \(y=x - 3\). But we also need to consider the grid. From the calculations, the intersection point is \((\frac{8}{3},-\frac{1}{3})\approx(2.67,-0.33)\). The region that satisfies both inequalities will be to the right of the intersection point (since the line \(y = x-3\) has a positive slope and \(y=-2x + 5\) has a negative slope) and in the area that is above \(y=-2x + 5\) and below \(y=x - 3\).

Answer:

The solution region is the area that is above the dashed line \(y = - 2x+5\) and below the solid line \(y=x - 3\), with the intersection point at \((\frac{8}{3},-\frac{1}{3})\) (or approximately \((2.67,-0.33)\)). To graph it, draw a dashed line for \(y=-2x + 5\), a solid line for \(y=x - 3\), shade above \(y=-2x + 5\) and below \(y=x - 3\), and the overlapping shaded region is the solution.