Solve Real-World Problems Using Inequalities

Solving real-world problems that involve inequalities is very much like solving problems that involve equations.

Example 1

cookie sale image from clipart In order to get a bonus this month, Leon must sell at least 120 newspaper subscriptions. He sold 85 subscriptions in the first three weeks of the month. How many subscriptions must Leon sell in the last week of the month?

Solution

Let x = the number of subscriptions Leon sells in the last week of the month. The total number of subscriptions for the month must be greater than 120, so we write :

85 + x ≥ 120.

We solve the inequality by subtracting 85 from both sides: x ≥ 35.

Leon must sell 35 or more subscriptions in the last week to get his bonus.

Check

To check the answer, we see that 85 + 35 = 120. If he sells 35 or more subscriptions, the total number of subscriptions he sells that month will be 120 or more. The answer checks out.

Example 2

Virenas Scout troop is trying to raise at least $650 this spring. How many boxes of cookies must they sell at $4.50 per box in order to reach their goal?

Solution

Let x = number of boxes sold. Then the inequality describing this problem is 4.50 ≥ 650.

We solve the inequality by dividing both sides by 4.50: x ≥ 144.44.

We round up the answer to 145 since only whole boxes can be sold.

Virenas troop must sell at least 145 boxes.

Check

If we multiply 145 by $4.50 we obtain $652.50, so if Virenas troop sells more than 145 boxes they will raise more than $650. But if they sell 144 boxes, they will only raise $648,

which is not enough. So they must indeed sell at least 145 boxes. The answer checks out.

Example 3

The width of a rectangle is 20 inches. What must the length be if the perimeter is at least 180 inches?

Solution

Let x = length of the rectangle. The formula for perimeter is:

Perimeter = 2 x length + 2 x width

Since the perimeter must be at least 180 inches, we have 2x + 2(20) ≥ 180.

Simplify: 2x + 40 ≥ 180

Subtract 40 from both sides: 2x ≥ 140

Divide both sides by 2: x ≥ 70

The length must be at least 70 inches.

Check

If the length is at least 70 inches and the width is 20 inches, then the perimeter is at least 2(70)+2(20) =180 inches. The answer checks out.

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