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Addition/Subtraction Property of Inequalities

Topic: Time Value of MoneyChapter: Chapter 3
If $a > b$, then $a + c > b + c$ and $a - c > b - c$. This property also applies to $<, \ge, \le$.
This rule states that adding or subtracting the same number to both sides of an inequality does not change the direction of the inequality sign. In TVM applications, this might be used when isolating variables after an initial calculation, for example, when rearranging an expression like (1+r)n−1>X(1+r)^n - 1 > X to find the range for (1+r)n(1+r)^n.

Multiplication/Division Property by a Positive Number

Topic: Time Value of MoneyChapter: Chapter 3
If $a > b$ and $c > 0$, then $ac > bc$ and $\frac{a}{c} > \frac{b}{c}$. This property also applies to $<, \ge, \le$.
Multiplying or dividing both sides of an inequality by a positive number does not change the direction of the inequality sign. This is commonly used in TVM problems when scaling an investment or return, for instance, simplifying an inequality like PV⋅(1+r)n≥FVtargetPV \cdot (1+r)^n \ge FV_{\text{target}} by dividing both sides by the positive principal value PVPV.

Multiplication/Division Property by a Negative Number

Topic: Time Value of MoneyChapter: Chapter 3
If $a > b$ and $c < 0$, then $ac < bc$ and $\frac{a}{c} < \frac{b}{c}$. The inequality sign is reversed.
This is a critical rule: multiplying or dividing both sides of an inequality by a negative number reverses the direction of the inequality sign. While less common in direct TVM formula manipulation (as interest rates, time, and principal are usually positive), it's essential when dealing with variables that might take negative values or when rearranging complex expressions where a negative coefficient is involved.

Solving a Simple Linear Inequality

Topic: Time Value of MoneyChapter: Chapter 3
To solve $ax + b \text{ op } c$, where $\text{op}$ is an inequality operator (e.g., $>, <, \ge, \le$), isolate $x$ using the properties above. For example: $ax > c - b \implies x > \frac{c-b}{a}$ (if $a > 0$).
This describes the general process of finding the range of values for an unknown variable (like an interest rate or time period) that satisfies a given financial condition expressed as an inequality. For instance, determining the minimum interest rate required for an investment to reach a certain future value or the maximum time period for a loan to not exceed a certain repayment amount.

General Form of a Linear Inequality (One Variable)

Topic: Solving Linear InequalitiesChapter: Chapter 3
ax+b<0 or ax+b>0 or ax+b≤0 or ax+b≥0ax + b < 0 \text{ or } ax + b > 0 \text{ or } ax + b \le 0 \text{ or } ax + b \ge 0
This represents the general form of a linear inequality with one variable 'x', where 'a' and 'b' are real numbers and 'a' is not equal to zero. The goal is to find the values of 'x' that satisfy this condition.

Rule for Addition/Subtraction

Topic: Solving Linear InequalitiesChapter: Chapter 3
If a<b, then a+c<b+c and a−c<b−c\text{If } a < b, \text{ then } a + c < b + c \text{ and } a - c < b - c
Adding or subtracting the same real number 'c' to both sides of an inequality does not change the direction of the inequality sign. This rule applies to all four inequality symbols (<, >, \le, \ge).

Rule for Multiplication/Division by a Positive Number

Topic: Solving Linear InequalitiesChapter: Chapter 3
If a<b and c>0, then ac<bc and ac<bc\text{If } a < b \text{ and } c > 0, \text{ then } ac < bc \text{ and } \frac{a}{c} < \frac{b}{c}
Multiplying or dividing both sides of an inequality by the same positive real number 'c' does not change the direction of the inequality sign. This rule applies to all four inequality symbols.

Rule for Multiplication/Division by a Negative Number

Topic: Solving Linear InequalitiesChapter: Chapter 3
If a<b and c<0, then ac>bc and ac>bc\text{If } a < b \text{ and } c < 0, \text{ then } ac > bc \text{ and } \frac{a}{c} > \frac{b}{c}
Multiplying or dividing both sides of an inequality by the same negative real number 'c' reverses the direction of the inequality sign. This is a critical rule to remember when solving inequalities.

Transitivity Property

Topic: Solving Linear InequalitiesChapter: Chapter 3
If a<b and b<c, then a<c\text{If } a < b \text{ and } b < c, \text{ then } a < c
This property states that if one quantity is less than a second, and the second is less than a third, then the first is also less than the third. This applies to all inequality symbols.

Basic Inequality Symbols

Topic: Logical ReasoningChapter: Chapter 3
a<b, a>b, a≤b, a≥ba < b, \ a > b, \ a \le b, \ a \ge b
These symbols represent 'less than', 'greater than', 'less than or equal to', and 'greater than or equal to' respectively. They define the relationship between two quantities 'a' and 'b'.

Addition/Subtraction Rule

Topic: Logical ReasoningChapter: Chapter 3
If a<b, then a+c<b+c and a−c<b−cIf \ a < b, \ then \ a + c < b + c \text{ and } a - c < b - c
Adding or subtracting the same real number 'c' to both sides of an inequality does not change the direction of the inequality sign. This rule applies to all inequality types (>, \le, \ge) as well.

Multiplication/Division by a Positive Number

Topic: Logical ReasoningChapter: Chapter 3
If a<b and c>0, then ac<bc and ac<bcIf \ a < b \text{ and } c > 0, \ then \ ac < bc \text{ and } \frac{a}{c} < \frac{b}{c}
Multiplying or dividing both sides of an inequality by the same positive real number 'c' does not change the direction of the inequality sign. This rule applies to all inequality types (>, \le, \ge) as well.

Multiplication/Division by a Negative Number

Topic: Logical ReasoningChapter: Chapter 3
If a<b and c<0, then ac>bc and ac>bcIf \ a < b \text{ and } c < 0, \ then \ ac > bc \text{ and } \frac{a}{c} > \frac{b}{c}
Multiplying or dividing both sides of an inequality by the same negative real number 'c' reverses the direction of the inequality sign. This is a critical rule to remember for all inequality types.

Transitive Property of Inequality

Topic: Logical ReasoningChapter: Chapter 3
If a<b and b<c, then a<cIf \ a < b \text{ and } b < c, \ then \ a < c
This property states that if one quantity is less than a second quantity, and the second quantity is less than a third, then the first quantity is also less than the third. This applies similarly to 'greater than' inequalities.

Solving a General Linear Inequality

Topic: Logical ReasoningChapter: Chapter 3
ax+b op c  ⟹  ax op c−b  ⟹  x op’ c−baax + b \text{ op } c \implies ax \text{ op } c - b \implies x \text{ op' } \frac{c - b}{a}
To solve a linear inequality, isolate the variable 'x' using algebraic operations similar to solving linear equations. Remember to apply the rules for changing the inequality sign, especially when multiplying or dividing by a negative number. 'op' represents any inequality operator (<, >, \le, \ge), and 'op'' represents the potentially reversed operator.
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