HOW TO GRAPH LINEAR INEQUALITIES IN TWO VARIABLES

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

Problem 1 :

y < -3/5x - 7

Solution :

y = -3/5x - 7

x -intercept : y = 0

-3/5x - 7 = 0

-3/5x = 7

x = -35/3

x -intercept:  (-35/3, 0)

y -intercept : x = 0

y = -3/5(0) - 7

y = -7

y -intercept:  (0, -7)

Check :

Problem 2 :

3x + 2y ≥ 7

Solution :

3x + 2y = 7

2y = -3x + 7

 y = -3/2x + 7/2

x -intercept : y = 0

-3/2x + 7/2 = 0

3/2x = 7/2

x = 7/3

x -intercept:  (7/3, 0)

y -intercept : x = 0

y = -3/2x + 7/2

y = -3/2(0) + 7/2

y = 7/2

y -intercept:  (0, 7/2)

Check :

Problem 3 :

-4x + 2y < 3

Solution :

-4x + 2y = 3

2y = 4x + 3

 y = 2x + 3/2

x -intercept : y = 0

2x + 3/2 = 0

2x = -3/2

x = -3/4

x -intercept:  (-3/4, 0) 

y -intercept : x = 0

y = 2x + 3/2

y = 2(0) + 3/2

y = 3/2

y -intercept:  (0, 3/2)

Check :

Problem 4 :

Given x + y > 2

y ≤ 3x - 2

Which graph shows the solutions of the given set of inequalities ?

Solution :

x + y > 2

y ≤ 3x - 2

x + y = 2

Choosing one of the points above the line (2, 2), 

x + y > 2

applying it, 2 + 2 > 2 ==> 4 > 2 (true)

So, we may shade the region above the line.

y = 3x - 2

Choosing one of the points below the line (2, -2), 

y ≤ 3x - 2

-2 ≤ 3(-2) - 2

-2 ≤ -8 (true)

So, we may shade the region above the line.

So, option 2 is correct.

Problem 5 :

An online store sells digital cameras and cell phones. The store makes a $100 profit on the sale of each digital camera x and a $50 profit on the sale of each cell phone y. The store wants to make a profit of at least $300 from its sales of digital cameras and cell phones. Write and graph an inequality that represents how many digital cameras and cell phones they must sell. Identify and interpret two solutions of the inequality.

Solution :

100x + 50y ≥ 300

50y ≥ -100x + 300

Dividing by 50 on both sides, we get

y ≥ (-100x/50) + (300/50)

y ≥ -2x + 6

Let y = -2x + 6

x - intercept :

Put y = 0

-2x + 6 = 0

-2x = -6

x = 3

(3, 0)

y - intercept :

Put x = 0

y = -2(0) + 6

y = 6

(0, 6)

It must be a solid line.

Choosing one of the point above the line, we get (2, 6).

6 ≥ -2(2) + 6

6 ≥ -4 + 6

6 ≥ 2

True

Since it is true, we shade the region above the line.

Problem 6 :

Write an inequality that represents the graph.

Solution :

Since it is dotted line, we may use the less than sign or greater than sign.

Choosing two points which lies on the line (2, -1) and (0, -2)

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

= -1/(-2)

= 1/2

Equation of the line will be in the form y = mx + b

y = (1/2) x + (-2)

Let us fix the less than sign and choose the point from the solution region.

y < (1/2) x - 2

Choosing the point (2, 2) from the solution region, we get

2 < (1/2) (2) - 2

2 < 1 - 2

2 < -1

False

Choosing the greater than sign, 

y > (1/2) x - 2

Applying the point (2, 2), we get

2 > (1/2) (2) - 2

2 > 1 - 2

2 > -1

True

So, the inequality represented by the graph is y > (1/2) x - 2.