Mastering Inequalities for Competitive Exams (CAT Quant)

Welcome to the foundational module on Inequalities, a crucial topic for excelling in the Quantitative Aptitude section of competitive exams like the CAT. Inequalities are mathematical statements that compare two values or expressions using symbols like < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). Understanding these concepts is key to solving a wide range of problems, from number systems to functions and optimization.

Basic Concepts and Symbols

At its core, an inequality establishes a relationship between two quantities that are not necessarily equal. The primary symbols used are:

Symbol	Meaning	Example
<	Less Than	3 < 5 (3 is less than 5)
	Greater Than	7 > 2 (7 is greater than 2)
≤	Less Than or Equal To	x ≤ 10 (x can be 10 or any number less than 10)
≥	Greater Than or Equal To	y ≥ 5 (y can be 5 or any number greater than 5)

Properties of Inequalities

Understanding how inequalities behave when manipulated is vital for solving them. These properties are similar to those of equations, with one critical difference when multiplying or dividing by a negative number.

Adding or subtracting the same number to both sides of an inequality does not change its direction.

If a < b, then a + c < b + c and a - c < b - c. This is a fundamental rule for isolating variables.

Consider an inequality like $x - 3 < 5$ . To isolate $x$ , we add 3 to both sides: $(x - 3) + 3 < 5 + 3$ , which simplifies to $x < 8$ . The direction of the inequality symbol (<) remains unchanged.

Multiplying or dividing both sides of an inequality by a positive number does not change its direction.

If a < b and c > 0, then ac < bc and a/c < b/c. This property allows scaling of inequalities.

For example, if $2x < 10$ , dividing both sides by 2 (a positive number) gives $x < 5$ . The inequality direction stays the same.

Multiplying or dividing both sides of an inequality by a negative number reverses its direction.

If a < b and c < 0, then ac > bc and a/c > b/c. This is the most common pitfall when solving inequalities.

Take the inequality $-3x < 12$ . If we divide both sides by -3, we must reverse the inequality sign: $x > -4$ . Failing to do this leads to an incorrect solution.

What happens to the inequality sign if you multiply both sides by -2?

The inequality sign reverses.

Solving Linear Inequalities

Solving linear inequalities involves applying the properties discussed to isolate the variable. The goal is to find the range of values for the variable that satisfy the given inequality.

Consider the inequality $3x - 5 \ge 7$ . To solve this, we first add 5 to both sides: $3x \ge 12$ . Then, we divide both sides by 3 (a positive number), so the inequality direction remains the same: $x \ge 4$ . This means any value of $x$ that is 4 or greater will satisfy the original inequality. The solution set can be visualized on a number line, with a closed circle at 4 and an arrow extending to the right.

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Solve for x:

2x + 3 < 11

Subtract 3 from both sides: $2x < 8$ . Divide by 2: $x < 4$ .

Compound Inequalities

Compound inequalities involve two or more inequalities joined by 'and' or 'or'. They represent a range of values that satisfy all conditions (for 'and') or at least one condition (for 'or').

For example, $-2 < x + 1 \le 5$ is a compound inequality. To solve it, we apply operations to all three parts simultaneously to isolate $x$ in the middle. Subtract 1 from all parts: $-2 - 1 < x \le 5 - 1$ , which simplifies to $-3 < x \le 4$ . This means $x$ can be any value greater than -3 and less than or equal to 4.

What does the compound inequality

x < 5

AND

x > 2

represent?

It represents values of $x$ that are strictly between 2 and 5 (i.e., $2 < x < 5$ ).

Inequalities with Absolute Values

Absolute value inequalities involve the absolute value function, which represents the distance of a number from zero. Key rules to remember are:

If $|x| < a$ (where $a > 0$ ), then $-a < x < a$ . If $|x| > a$ (where $a > 0$ ), then $x < -a$ OR $x > a$ .

For instance, to solve $|2x - 1| < 5$ , we rewrite it as $-5 < 2x - 1 < 5$ . Adding 1 to all parts gives $-4 < 2x < 6$ . Dividing by 2 yields $-2 < x < 3$ .

How would you rewrite the inequality

|x| > 3

as two separate linear inequalities?

$x < -3$ or $x > 3$ .

Quadratic Inequalities

Quadratic inequalities involve expressions with a squared term, like $ax^2 + bx + c$ . Solving these typically involves finding the roots of the corresponding quadratic equation and then testing intervals on a number line.

For $x^2 - 5x + 6 > 0$ , first find the roots of $x^2 - 5x + 6 = 0$ , which are $x=2$ and $x=3$ . These roots divide the number line into three intervals: $(-\infty, 2)$ , $(2, 3)$ , and $(3, \infty)$ . Test a value from each interval in the original inequality to determine which intervals satisfy it. For example, testing $x=0$ gives $6 > 0$ (True), so $(-\infty, 2)$ is part of the solution. Testing $x=2.5$ gives $6.25 - 12.5 + 6 = -0.25 > 0$ (False), so $(2, 3)$ is not part of the solution. Testing $x=4$ gives $16 - 20 + 6 = 2 > 0$ (True), so $(3, \infty)$ is part of the solution. The solution is $x < 2$ or $x > 3$ .

What are the critical points for solving the quadratic inequality

x^2 - 4 \le 0

The roots of $x^2 - 4 = 0$ , which are $x = -2$ and $x = 2$ .

Key Takeaways for Competitive Exams

When tackling inequalities in exams, remember to: 1. Pay close attention to the inequality symbol. 2. Always reverse the sign when multiplying or dividing by a negative number. 3. For compound inequalities, apply operations to all parts. 4. For absolute value inequalities, break them down into cases or use the distance concept. 5. For quadratic inequalities, use roots and test intervals.

Learning Resources

Inequalities - Concepts and Formulas(documentation)

Provides a concise overview of inequality concepts, properties, and common formulas relevant for competitive exams.

Introduction to Inequalities(tutorial)

A series of video lessons and practice exercises covering the basics of solving and graphing linear inequalities.

Solving Inequalities - CAT Preparation(blog)

A blog post specifically tailored for CAT aspirants, explaining common types of inequality problems and strategies to solve them.

Absolute Value Inequalities(tutorial)

Focuses on understanding and solving inequalities involving absolute values, a common challenge in quantitative aptitude tests.

Quadratic Inequalities Explained(documentation)

A clear explanation of how to solve quadratic inequalities, including the method of testing intervals on a number line.

CAT Quantitative Aptitude: Inequalities(video)

A video tutorial demonstrating how to approach and solve various types of inequality problems typically found in the CAT exam.

Properties of Inequalities(documentation)

Details the fundamental properties of inequalities and how they are applied in solving algebraic problems.

Inequalities - Practice Problems for CAT(tutorial)

Offers a set of practice questions with solutions for inequalities, allowing learners to test their understanding and problem-solving skills.

Understanding Compound Inequalities(documentation)

Explains the concept of compound inequalities and provides methods for solving them, including graphical representations.

Inequalities - Wikipedia(wikipedia)

A comprehensive overview of mathematical inequalities, their history, and various types, providing a broader theoretical context.