On a coordinate plane, 2 straight lines are shown. The first solid line has a positive slope and goes through (0, negative 2) and (2, 0). Everything to the left of the line is shaded. The second dashed line has a negative slope and goes through (0, 2) and (4, 0). Everything below and to the left of the line is shaded.

Question

Which system of linear inequalities is represented by the graph?

y > x – 2 and x – 2y < 4
y > x + 2 and x + 2y < 4
y > x – 2 and x + 2y < 4
y > x – 2 and x + 2y < –4

Answer 1

To determine the correct system of linear inequalities represented by the graph, we need to analyze the two lines and the shaded regions.

1. First line (solid line):

   • It connects the points (0, -2) and (2, 0).
   • The slope of this line can be calculated: rise/run = (0 - (-2)) / (2 - 0) = 2/2 = 1.
   • The equation of the line can be found using point-slope form or slope-intercept form. From the points, we can see two important points:
     • When x = 0, y = -2 => (0, -2)
     • When x = 2, y = 0 => (2, 0)
   • The equation in slope-intercept form (y = mx + b) yields: $$ y = x - 2 $$
   • Since everything to the left of the line is shaded, the inequality represented is: $$ y > x - 2 $$

2. Second line (dashed line):

   • It connects the points (0, 2) and (4, 0).
   • The slope is negative: rise/run = (0 - 2) / (4 - 0) = -2/4 = -1/2.
   • The equation of this line can also be derived: $$ y = -\frac{1}{2}x + 2 $$
   • To convert this to a standard form: $$ x + 2y = 4 $$
   • Since everything below and to the left of the line is shaded, we use the strict inequality for dashed lines: $$ x + 2y < 4 $$

Combining these two inequalities, we find the correct system of inequalities:

• \( y > x - 2 \)
• \( x + 2y < 4 \)

Now, looking at the options:

• y > x – 2 and x + 2y < 4 matches our derived inequalities.

Thus, the correct answer is:

y > x - 2 and x + 2y < 4

Answer 2

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