GRAPHING INEQUALITIES IN TWO VARIABLES

Graph the solutions to each of the following inequalities on a separate set of axes.

Problem 1 :

y ≤ 3x + 1

Solution :

y = 3x + 1

x -intercept : y = 0

3x + 1 = 0

3x = -1

x = -1/3

x -intercept: (-1/3, 0)

y -intercept : x = 0

y = 3(0) + 1

y = 0 + 1

y = 1

y -intercept: (0, 1)

Check :

So, we shade the region below the line.

Problem 2 :

y ≥ -2x + 3

Solution :

y = -2x + 3

x -intercept : y = 0

-2x + 3 = 0

-2x = -3

x = 3/2

x -intercept: (3/2, 0)

y -intercept : x = 0

y = -2(0) + 3

y = 3

y -intercept:  (0, 3)

Check :

Problem 3 :

y > 4x - 2

Solution :

y = 4x - 2

x -intercept : y = 0

4x - 2 = 0

4x = 2

x = 1/2

x -intercept: (1/2, 0)

y -intercept : x = 0

y = 4(0) - 2

y = -2

y -intercept:  (0, -2)

Check :

Problem 4 :

y < -3x - 5

Solution :

y = -3x - 5

x -intercept : y = 0

-3x - 5 = 0

-3x = 5

x = -5/3

x -intercept: (-5/3, 0)

y -intercept : x = 0

y = -3(0) - 5

y = -5

y -intercept is (0, -5)

Check :

Problem 5 :

y ≤ 3

Solution :

y = 3

y -intercept is (0, 3)

Problem 6 :

x > 1

Solution :

x = 1

x -intercept:  (1, 0)

Problem 7 :

y > 2/3x + 8

Solution :

y = 2/3x + 8

x -intercept : y = 0

2/3x + 8 = 0

2/3x = -8

x = -12

x -intercept:  (-12, 0)

y -intercept : x = 0

y = 2/3(0) + 8

y = 8

x -intercept:  (0, 8)

Check :

Problem 8 :

You have two part-time summer jobs, one that pays $9 an hour and another that pays $12 an hour. You would like to earn at least $240 a week. Write an inequality describing the possible amounts of time you can schedule at both jobs.

Solution :

Let x be the number of hours working for the first part time job

Let y be the number of hours working for the second part time job.

9x + 12y ≥ 240

12y ≥ -9x + 240

y ≥ (-9x + 240)/12

y ≥ (-9x/12) + (240/12)

y ≥ (-3x/4) + 20

Finding x and y-intercepts :

x-intercept, put y = 0

0 = (-3x/4) + 20

-20 = -3x/4

80/3 = x

x = 26.6

y-intercept, put x = 0

y = 0 + 20

y = 20

Problem 9 :

The graph of an inequality is shown below.

a) Write the inequality represented by the graph.

b) On the same set of axes, graph the inequality x + 2y < 4

c) The two inequalities graphed on the set of axes form a system. Oscar thinks that  (2, 1) the point is in the solution set for this system of inequalities. Determine and state whether you agree with Oscar. Explain your reasoning.

Solution :

Choosing two points lies on the line, we get (2, 1) and (0, -3)

slope m = (-3 - 1) / (0 - 2)

= -4/(-2)

= 2

Since it is solid line, we have to use ≤ or ≥ sign.

y = 2x + (-3)

y = 2x - 3

So, the inequality represented by the graph is y ≥ 2x - 3.

x + 2y < 4

2y = -x + 4

y = (-1/2)x + 4

(8, 0) and (0, 4)

Choosing the point above the line (4, 8),

 4 + 2(8) < 4

4 + 16 < 4

20 < 4

False, so we have to shade the region below the line.

(2, 1) is the point of intersection of the inequality. So, it must be a solution.