Solve the following systems of linear inequalities

x+2y is less than or equal to 6

y is greater than -4x

To solve the system of linear inequalities:

1. Let's start with the first inequality:
x + 2y ≤ 6

We can rewrite it as:
2y ≤ -x + 6

Then divide the entire inequality by 2:
y ≤ (-1/2)x + 3

2. Now let's move on to the second inequality:
y > -4x

This inequality is already in slope-intercept form (y > mx), so no further manipulation is needed.

3. Graph the system of inequalities on a coordinate plane:
Start by plotting the line y = (-1/2)x + 3, which has a slope of -1/2 and a y-intercept of 3. The line will have a negative slope and is a solid line since it includes the "less than or equal to" part.

Next, draw a dashed line for y = -4x, which has a slope of -4 and goes through the origin (0,0).

Finally, shade the region above the dashed line y = -4x since the inequality states y is greater than -4x. Shade the region below and including the solid line y = (-1/2)x + 3 since the inequality states y is less than or equal to (-1/2)x + 3.

4. The shaded region where the two regions overlap represents the solution to the system of inequalities.

Here is a rough sketch of the graph:

To solve the system of linear inequalities, follow these steps:

Step 1: Graph the first linear inequality. To do this, first change the inequality sign to an equal sign to draw the line. Rewrite the inequality as an equation: x + 2y = 6. Next, choose two arbitrary values for x and solve for y to plot the line accurately. For example, when x = 0, y = 3, and when x = 6, y = 0. Plot these two points on the graph and draw a straight line passing through them.

Step 2: Shade the region that satisfies the inequality. Choose a test point that is not on the line you just graphed. For example, (0,0) is a good test point. Substitute the test point's coordinates into the original inequality. If the inequality holds true, shade the region containing the test point. If the inequality is false, shade the region on the opposite side of the line. In this case, since (0,0) satisfies the inequality x + 2y ≤ 6, shade the region below the line.

Step 3: Graph the second linear inequality. Change the inequality sign to an equal sign: y = -4x. Choose two arbitrary values for x and solve for y to plot the line accurately. For example, when x = 0, y = 0, and when x = 1, y = -4. Plot these two points on the graph and draw a straight line passing through them.

Step 4: Shade the region that satisfies the inequality. Choose another test point, such as (0,0), and substitute it into the original inequality. If the inequality is true, shade the region containing the test point. Otherwise, shade the region on the opposite side of the line. In this case, since (0,0) satisfies the inequality y > -4x, shade the region above the line.

Step 5: Find the intersection region. The solution to the system of linear inequalities is the region where the shaded regions from both inequalities overlap. This region represents the set of values that satisfy both inequalities in the system.

To summarize: Graph the individual linear inequalities, shade the regions that satisfy each inequality, and find the intersection region where the shaded regions overlap.