SOLVING SYSTEMS OF LINEAR INEQUALITIES

Mr. Waleed wants to rent a car for a business trip. A car rental company advertises that their rental rate is Rs.9000 plus Rs.60 per km. Mr. Waleed would like to compare this rate with the rates offered by other car rental agencies. First, he determines the equation containing the points that represent the relationship between the numbers of km $(x)$ and the total cost of the rental $(y)$ . The initial cost of the car is Rs. 9000. Since this is the point where zero km are driven, it would be the y- intercept of the graph.

The slope would be the rate of change in the total cost. In this case, the rate is Rs.60 per km. Since the slope is 60. Thus, an equation of the line is $y = 60x + 9000$ .

The graph of $y = 60x + 9000$ separates the coordinate plane into two regions.

The line is called boundary of the region. To graph an inequality, first you graph the boundary and then determine which region to shade. The graph $y > 60x + 9000$ contains the points that are located above the boundary.

In that region, the value of the dependent variable $y$ is greater than the value of $60x + 9000$ . This graph represents car rental costs that are greater than those offered by car rental company A. For example, another company B charges 14000 for a car rental with 50km.

The point (50, 14000) lies above the boundary.

A linear inequality in two variables: such as $x - 3y < 6$ is the result of replacing the '=' in a linear equation with or $\geq$ . A solution of an inequality in two variables $x$ and $y$ is an ordered pair $(x,y)$ that produces a true statement when the values of $x$ and $y$ are satisfied into the inequality.

Example: Which ordered pair is not a solution of $x - 3y\leq 6?$

a. (0,0)

b. (6,-1)

c. (10,3)

d. (-1,2)

Solution:

Test (0,0): $x - 3y\leq 6$

$0 - 3(0)\leq 6$

$0\leq 6$ True

Test (6,-1): $x - 3y\leq 6$

$6 - 3(-1)\leq 6$

$9\leq 6$ False

Similarly (10,3) and (-1,2) also true but (6,-1) is not a solution.

Solving Systems of Linear Inequalities