To graph the solution to the system of inequalities on the coordinate plane, follow these steps:
### Inequality 1: [tex]\( y > 2x + 4 \)[/tex]
1. Graph the boundary line: Start by graphing the line [tex]\( y = 2x + 4 \)[/tex]. This is the boundary line.
- Slope: The slope is 2.
- Y-intercept: The y-intercept is 4.
- Plot the y-intercept at (0,4).
- Use the slope to find another point. From (0,4), go up 2 units and to the right 1 unit to find the point (1,6).
- Draw the line through these points. Since this is a strict inequality (y > 2x + 4), use a dashed line to represent [tex]\( y = 2x + 4 \)[/tex].
2. Shade the region: Since [tex]\( y > 2x + 4 \)[/tex], shade the region above the dashed line.
### Inequality 2: [tex]\( x + y \leq 6 \)[/tex]
1. Graph the boundary line: Now graph the line [tex]\( x + y = 6 \)[/tex].
- Solve for y: [tex]\( y = 6 - x \)[/tex].
- Find the intercepts: For the x-intercept, let y = 0, then x = 6 (so plot (6,0)). For the y-intercept, let x = 0, then y = 6 (so plot (0,6)).
- Draw the line through these points. Since this inequality is inclusive ([tex]\( x + y \leq 6 \)[/tex]), use a solid line to represent [tex]\( x + y = 6 \)[/tex].
2. Shade the region: Since [tex]\( x + y \leq 6 \)[/tex], shade the region below the line.
### Finding the Solution Region
The solution to the system of inequalities will be the region where the two shaded regions overlap. Look for the area on the graph where both the shading from the first inequality (above the dashed line) and the shading from the second inequality (below the solid line) combine.
### Summary of steps on the coordinate plane
1. Plot the dashed line [tex]\( y = 2x + 4 \)[/tex] and shade above it.
2. Plot the solid line [tex]\( y = 6 - x \)[/tex] and shade below it.
3. Highlight the overlapping shaded region, which is the solution to the system.
When graphing, make sure the area you shade reflects the accurate intersection of both inequalities.
Here's what to look for visually:
1. Dashed line through (0,4) and (1,6), region above the line shaded.
2. Solid line through (6,0) and (0,6), region below the line shaded.
3. The shaded region where both conditions are true is the solution.
These tools in the table can be used appropriately:
- Select tool to choose different points or lines.
- Line for drawing the boundary lines.
- Dashed Line for the boundary of [tex]\( y = 2x + 4 \)[/tex].
- Shaded Region to highlight the areas satisfying the inequalities.
