Solve the following linear inequality graphically.

x+2y<4
x-y<5

Each inequality is satisfied in a particular region of an x,y plot. The first inequality is satisfied below the line y = -x/2 + 2, for example.

The second inequality is satisfied above the line y = x + 5.

Draw both lines and shade in the areas that the inequalities represent. You will find that the two lines intersect at x=-2, y=3. A wedged shaped region to the left of that point satisfies both inequalities.

To solve the linear inequalities graphically, we will start by graphing each inequality on the x, y plane.

1. Graphing x + 2y < 4:
Start by rewriting the inequality in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.
x + 2y < 4 becomes y < (-1/2)x + 2
Plot the y-intercept at (0, 2) and use the slope of -1/2 to find an additional point (e.g., you can move down 1 and to the right 2 from the y-intercept).
Draw a dotted line to represent the inequality y = (-1/2)x + 2.
Shade the region below the line (since y is less than the equation).

2. Graphing x - y < 5:
Rewrite the inequality in slope-intercept form.
x - y < 5 becomes y > x - 5 (we change the inequality direction when multiplying/dividing by a negative number)
Plot the y-intercept at (0, -5) and use a slope of 1 to find an additional point (e.g., move up 1 and to the right 1 from the y-intercept).
Draw a dotted line to represent the inequality y = x - 5.
Shade the region above the line (since y is greater than the equation).

Now, we can identify the region that satisfies both inequalities by observing where the shaded areas overlap. In this case, we see that the lines intersect at x = -2 and y = 3. The shaded region to the left of this point satisfies both inequalities.

So, the solution to the system of inequalities is the region to the left of the intersection point (x = -2, y = 3), including the line itself.