Question

Solve the following system of inequalities graphically on the set of axes below. State the coordinates of a point in the solution set. Give a point where both shades meet and points in between also tell if the line need to be shaded or not and also give the points of where both inequality's should be graph'd

y, is less than, x, plus, 2
y<x+2
y, is less than, minus, one half, x, minus, 4
y<−
2
1
​
x−4
x
y
x
y
Answer
Attempt 1 out of 2

Point: left parenthesis(
,
right parenthesis)

Answer 1

The system is
y < x + 2
y < −(1/2)x − 4

Lines to draw
- y = x + 2: dashed (strict <). Two points: (0, 2) and (−2, 0).
- y = −(1/2)x − 4: dashed (strict <). Two points: (0, −4) and (2, −5).

Intersection of the two lines
Solve x + 2 = −(1/2)x − 4:
(3/2)x = −6 ⇒ x = −4, y = −2. So the lines meet at (−4, −2). This is where the two boundary lines cross; because the inequalities are “<” these boundary points are NOT included.

Shading / solution region
Shade the region below each dashed line. The solution set is the overlap: points that are below both lines. Example points in the solution set:
- (0, −5): check −5 < 0+2 (true) and −5 < −4 (true).
- (−4, −3): at x = −4 both lines give y = −2, and −3 < −2 (true).
- (2, −6): −6 < 4 (true) and −6 < −5 (true).

Point where both shades meet (boundary crossing): (−4, −2) — this point is on both boundary lines but is NOT part of the solution because the inequalities are strict.

Answer (one point in solution): (0, −5)