Linear inequalities and linear equations are widely used in the business world to compare relationships between total cost and total revenue and are also popular in calculating important information e.g. break even point, equilibrium point and elasticity co-efficient etc. The system of equations or inequalities provides an effective method for managers to quantify their cost and revenue variables that can be solved to reveal valuable information which can greatly assist in managerial decision making.

Example: Suppose a sewing machine business produces sewing machines and the manager wants to calculate the break even point for the number of machines that he produces monthly. In other words he wants to calculate the number of sewing machines that he would have to sell to break even, or a point where he would neither make profit nor incur any loss. Suppose the following information is available with the manager:

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Cost of manufacturing/unit= \$300;

Selling Price/unit= 400;

Number of units produced/month=40;

Number of machines to be sold per month to break even= X;

The equation relating total revenue and total cost at break even point is:

TR-TC=0……………………………………………………………………………… (1)

Where TR= Selling Price* Number of units sold and TC= Manufacturing Cost/Unit* Number of units produced per month.

Hence given on the above information and plugging the information in equation (1), the resultant equation is:

400X- (300*40) =0…………………………………………………………………… (2)

X= 12000/400= 30 Units are required to reach break even point.

Equation (1) can also be written in the form of an Inequality to make or cross the break even point.

TR>=TC……………………………………………………………………………… (3)

This means that in order to break even or cross the breakeven point TR should be greater than or equal to TC. Plugging values in inequality (2), we can find a relationship between TR and TC.

(400*X)>= (300*40)…………………………………………………………………. (4)

The power of an inequality is that it allows multiple values for the unknown variable which in this case is “X”. This is the major difference between a linear equation and a linear inequality. Linear inequalities can have multiple values for the unknown variable while linear equations can only have a single value for the required variable.

Solving the inequality (4) for X produces X>=30. This means that the manager of the sewing machine business will achieve or cross the break even point for any number of machines sold greater than or equal to 30.

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