WRITING SYSTEM OF INEQUALITIES FROM A GRAPH

Problem 1 :

What system of inequalities best represents the graph shown below?

A)  y > - 2 and y > x +1

B)  y > - 2 and y < x + 1

C)  y < - 2 and y > x +1

D)  y < - 2 and y < x + 1

Solution :

In the graph, we see one raising line and one horizontal line.

Raising line passes through the points (-1, 0) and (0, 1).

Slope of the raising line :

Slope (m) = (y₂ - y₁)/(x₂ - x₁)

= (1 - 0) / (0 + 1)

= 1/1

= 1

y-intercept = 1

Equation of the raising line :

y = mx + b

y = 1x + 1

y = x + 1

Equation of horizontal line :

y = -2

When we observe the shaded region, it is above -2. So it should be greater(>) than -2. We can clearly reject options C and D.

Selecting one of the points ex : (2, 1) from the shaded region and applying it.

So, option B is correct.

Problem 2 :

Which system of linear inequalities is represented by this graph?

A) y ≥ (1/2)x + 3 and y ≥ x - 2

B) y ≥ 2x + 3 and y ≤ x - 2

C) 2x - y ≥ 3 and x + y ≤ 2

D) 2x + y ≥ 3 and x - y ≥ 2

Solution :

In the given graph, we have one raising line and one falling line.

Raising line has -2 as y-intercept and it passes through (0, -2) and (3, 1).

Slope of raising line :

m = (y₂ - y₁)/(x₂ - x₁)

= (1 + 2) / (3 - 0)

= 3/3

= 1

y-intercept = -2

Equation of raising line :

y = mx + b 

y = 1x - 2

Any two points from the falling line are (0, 3) and (1, 1) and it has the y-intercept is 3.

Slope (m) = (1 - 3) / (1 - 0)

m = -2/1

m = -2

Equation of the falling line :

y = -2x + 3

By applying one of the points from the solution region in option D, we get

So, option D is correct.