BUS112: Introduction to Business Mathematics
Inequalities (Chapter 2.6)
Mohammad Alamgir Hossain, PhD
June 28, 2025
June 28, 2025
Properties of Inequalities
For real numbers a, b, and c,
(a) if a < b, then a + c < b + c,
(b) if a < b, and if c > 0, then ac < bc,
(c) if a < b, and if c < 0, then ac > bc.
Example
Consider the true statement, −3 < 5
(a) −3 + 2 < 5 + 2 =⇒ −1 < 7 ,
(b) −3 < 5 and 2 > 0, then −3 · 2 < 5 · 2 =⇒ −6 < 10,
(c) −3 < 5 and −4 < 0, then (−3) · (−4) > 5 · (−4) =⇒ 12 > −20.
June 28, 2025
Types of Brackets with Examples
Different Kinds of Brackets
Parentheses (also called Round or Open Brackets):
Example: (a + b)2
Curly Brackets (also called Brace Brackets):
Example: {x ∈ R : x > 0}
Square Brackets (also called Box or Closed Brackets):
Example: [1, 5]
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Example 1
Solve the inequality
−3x + 5 > −7.
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Example 2
Solve the inequality
4 − 3y ≤ 7 + 2y .
Write the solution in interval notation and set notation and graph the solution
on a number line.
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Example 3
Solve the inequality
−2 < 5 + 3m < 20.
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Quadratic Inequality
A quadratic inequality is an inequality that can be written in the form
ax 2 + bx + c < 0,
for real numbers a 6= 0, b, and c. The symbol < can be replaced with >, ≤, or
≥.
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Example 4
Solve the inequality
x 2 − x − 12 < 0.
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Example 5
Solve the inequality
2x 2 + 5x − 12 ≥ 0.
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Example 6
Solve the inequality
5
≥ 1.
x +4
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Example 7
Solve the inequality
2x − 1
< 5.
3x + 4
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Business Applications of Inequalities
Example: Budget Constraint
A business wants to produce x units of a product. The cost per unit is 120
Taka. The total production budget is at most 6000 Taka.
Inequality Model:
120x ≤ 6000 ⇒ x ≤ 50
Interpretation: The business can produce up to 50 units without exceeding
the budget.
Another Example: Minimum Sales Requirement
To remain profitable, a company must sell at least 300 units of a product
priced at 40 Taka.
Inequality:
40x ≥ 40 × 300 = 12, 000 ⇒ x ≥ 300
Interpretation: Sales must be 300 units or more to ensure profit.
June 28, 2025