In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. y>2x−3, y<−x+6

Verified step by step guidance

1. Step 1: Identify the inequalities in the system: $y>4x-2$ and $y<-3x+9$.
2. Step 2: Graph the boundary lines for each inequality. For $y=4x-2$, draw a line with slope 4 and y-intercept -2. For $y=-3x+9$, draw a line with slope -3 and y-intercept 9.
3. Step 3: Determine the shading for each inequality. For $y>4x-2$, shade the region above the line. For , shade the region below the line.
4. Step 4: Find the intersection of the two shaded regions. This overlapping area represents the solution set to the system of inequalities.
5. Step 5: Optionally, find the intersection point of the two boundary lines by solving the system of equations $y=4x-2$ and $y=-3x+9$ to better understand the boundaries of the solution region.

Key Concepts

Graphing Linear Inequalities

Graphing linear inequalities involves plotting the boundary line of the corresponding linear equation and then shading the region that satisfies the inequality. For '>', the boundary line is dashed, indicating points on the line are not included. The shaded area represents all points where the inequality holds true.

Slope-Intercept Form

The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept. This form helps quickly graph the line by starting at (0, b) and using the slope to find other points. Understanding this form is essential for accurately drawing the boundary lines of inequalities.

Solution Set of a System of Inequalities

The solution set of a system of inequalities is the region where the shaded areas of all inequalities overlap. It represents all points that satisfy every inequality simultaneously. If no such region exists, the system has no solution.