14. Inequality and Order-Based Comparison

Why This Chapter Matters

This chapter is not just about symbols like greater than and less than. In banking mocks, comparison questions often blend direct inequalities, linear equations, and quick quadratic-root logic.

Core Ideas

• Write the relation chain in order instead of trying to hold it mentally.
• When multiplying or dividing by a negative number, reverse the inequality sign.
• Statements such as greater than, less than, at least, and at most must be translated precisely.
• In quadratic comparison questions, first decide whether the roots are positive, negative, or mixed before doing any detailed factor work.
• If every possible value of quantity I lies above every possible value of quantity II, the relation is fixed even before exact solving.
• Some comparison questions are intentionally designed to end in "no definite relation". Recognising that quickly saves time.

High-Value Formulas

How To Approach Questions

1. Convert the verbal statement into symbols first.
2. Arrange the chain from left to right.
3. Apply sign-reversal carefully only when a negative factor is involved.
4. If two equations are being compared, decide whether sign analysis alone settles the relation before solving fully.
5. When both quantities have multiple possible values, test whether the relation stays fixed across all valid cases.

Worked Examples

Example 1

Prompt: If $a>b>c$, compare $a$ and $c$.

Approach: By transitivity, $a>c$.

Example 2

Prompt: If $x<y$, compare $-x$ and $-y$.

Approach: Multiplying by $-1$ reverses the sign, so $-x>-y$.

Example 3

Prompt: Compare the roots of $x^2-9x+20=0$ and $y^2+5y+6=0$.

Approach: The first equation factors to $(x-4)(x-5)=0$, so every possible $x$ value is positive. The second becomes $(y+2)(y+3)=0$, so every possible $y$ value is negative. Therefore $x>y$ always.

Example 4

Prompt: If $m>n$ and both are multiplied by $-3$, what happens to the inequality?

Approach: Multiplying by a negative number reverses the sign, so $-3m<-3n$.

Example 5

Prompt: Compare any root of $x^2-7x+10=0$ with any root of $y^2-5y+6=0$.

Approach: The roots of the first are $2,5$ and the roots of the second are $2,3$. Since equality occurs at $2$ but larger values also occur for $x$, no single strict relation like $x>y$ or $x<y$ is always true.

Common Mistakes

• Forgetting sign reversal after multiplying or dividing by a negative number.
• Assuming the bigger coefficient gives the bigger root.
• Mixing strict and non-strict inequalities without reading the condition.
• Answering from intuition instead of the actual relation chain.
• Choosing a strict relation when equality is also possible in some valid cases.

Quick Revision

Keep the chain visible, respect sign reversal, and use root-sign logic when equation comparison shows up.