Conquering math inequality questions especially quadratic equation questions require a systematic approach and a solid understanding of the concepts involved. Inequality questions assess a candidate's basic numeracy skills. Banks need employees who can work with numbers accurately and efficiently, as financial transactions and calculations are an integral part of banking operations.

Types of Inequalities You Need to Know: In bank exams and other competitive exams, you may encounter various types of inequalities. Here are the key types of inequalities you need to be familiar with:
  • Linear Inequalities: These are inequalities involving linear expressions, such as ax + b < c or mx - ny ≥ p. To solve linear inequalities, you perform operations like addition, subtraction, multiplication, and division on both sides while considering the direction of the inequality symbol.
  • Quadratic Inequalities: These inequalities involve quadratic expressions, such as ax^2 + bx + c < 0 or mx^2 - nx + p ≥ 0. Solving quadratic equation questions typically involves finding the roots of the associated quadratic equation questions and analyzing the sign of the expression within certain intervals.
  • Absolute Value Inequalities: These inequalities include absolute value expressions, such as |x - a| < b or |y + c| ≥ d. solving absolute value inequalities often requires considering two cases: one where the expression inside the absolute value is positive and another where it is negative.



Essential Inequality Symbols and Notations: In mathematics, inequality symbols and notations are used to represent relationships between quantities that are not equal to each other. Here are some essential inequality symbols and notations:
  • Greater Than (>), Less Than (<): (>) represents that one quantity is larger than another. For example, y > 2 means "y is greater than 2." (<) represents that one quantity is smaller than another. For example, x < 5 means "x is less than 5."
  • Greater Than or Equal To (≥), Less Tha36+n or Equal To (≤): The (≥) indicates that one quantity is either larger than or equal to another. For example, b ≥ 3 means "b is greater than or equal to 3." Whereas (≤) indicates that one quantity is either smaller than or equal to another. For example, a ≤ 7 means "a is less than or equal to 7."
  • Not Equal To (≠): Signifies that two quantities are not equal. For example, c ≠ 0 means "c is not equal to 0."

Solving Linear Inequalities

Solving linear inequalities is an important skill for tackling mathematics inequality questions, especially in bank exams. Let’s dive more and discuss it in detail:
  • Solving Inequalities with Addition and Subtraction: Your goal is to isolate the variable (usually denoted as 'x') on one side of the inequality sign. Achieve this by performing inverse operations (addition or subtraction) to move constants away from the variable.
  • Addition: If the variable is being subtracted from a constant, add the same value to both sides.
  • Subtraction: If the variable is being added to a constant, subtract the same value from both sides.
  • Solving Inequalities with Multiplication and Division: Solving inequalities with multiplication and division is another essential skill for conquering math inequality questions on bank exams. These problems involve expressions with variables and inequality symbols like <, >, ≤, or ≥.
  • Your objective is to isolate the variable (usually represented as 'x') on one side of the inequality sign. To do this, apply inverse operations (multiplication or division) to move constants away from the variable.
  • Multiplication: If the variable is being divided by a constant, multiply both sides by the reciprocal of that constant.
  • Division: If the variable is being multiplied by a constant, divide both sides by that constant.

Graphical Representation of Inequalities

Graphical representation of inequalities is a visual way to represent and understand the solution set of an inequality on a coordinate plane. This technique is particularly helpful when dealing with linear inequalities in two variables (usually denoted as x and y). Let’s explore more:
  • Plotting on a Number Line: Plotting on a number line is a useful technique for solving mathematical inequalities, which are often encountered in bank exams. It helps you visualize the solution set of an inequality and identify the valid range of values for the variable(s).
  • Shading Regions and Interval Notation: Based on the inequality type and the location of the critical points, shade the region on the number line that represents the valid solutions.
  • For < or >, shade the region to the left (for <) or right (for >) of the critical point(s). For ≤ or ≥, shade the region to the left (for ≤) or right (for ≥) of the critical point(s) and include the critical point(s) itself.

Solving Quadratic Inequalities

Quadratic inequalities involve expressions with quadratic terms (e.g., x^2) and inequality symbols (e.g., <, >, ≤, or ≥). Here's a step-by-step guide to solving quadratic inequalities:
  • Factoring Quadratic Inequalities: Try to factor the quadratic expression on the left-hand side (LHS) if it's factorable. This can help you find the critical points where the expression equals zero. For example, if you have (x - 2)(x + 3) < 0, you can determine that x = 2 and x = -3 are critical points.
  • Using the Quadratic Formula: Using the quadratic formula is a valuable technique for solving quadratic inequalities, and it can help you conquer math inequality questions on bank exams. Quadratic inequalities involve expressions with quadratic terms (e.g., x^2) and inequality symbols (e.g., <, >, ≤, or ≥).
  • Graphical Approach for Quadratic Inequalities: Rewrite the inequality in standard form by moving all terms to one side of the inequality sign, so it equals zero. This results in a quadratic inequality in the form of ax^2 + bx + c < 0 (or >, ≤, or ≥).
  • Treat the quadratic inequality as an equation and graph it on a coordinate plane. This graph will typically be a parabola. Determine whether the inequality symbol point’s upward (∪) or downward (∩) based on the inequality sign.

Working with Absolute Value Inequalities

Working with absolute value inequalities involves solving inequalities that contain absolute value expressions. Absolute value is represented by vertical bars (| |) and it represents the distance of a number from zero on the number line. Absolute value inequalities can have one of the following forms:
  1.   |x| < a
  2.   |x| ≤ a
  3.   |x| > a
  4.   |x| ≥ a
  • Solving Absolute Value Inequalities Algebraically:  Solving absolute value inequalities algebraically involves finding the values of the variable that satisfy the inequality by manipulating and simplifying algebraic expressions.
  • Graphical Interpretation of Absolute Value Inequalities: Graph the lines representing the boundaries defined by the absolute value inequalities.
  • For |x| < a or |x| ≤ a: Draw vertical lines at x = -a and x = a. These represent the boundary points where the absolute value inequality transitions from true to false or vice versa.
  • For |x| > a or |x| ≥ a: Draw vertical lines at x = -a and x = a. These represent the boundary points where the absolute value inequality transitions from false to true or vice versa.
  • Graphically interpreting absolute value inequalities provides a clear visual representation of the solution set, making it easier to understand the valid values of the variable(s) that satisfy the inequality.

Common Mistakes to Avoid When Dealing with Inequalities

Dealing with inequalities in mathematics requires attention to detail and precision. Here are some common mistakes to avoid when working with inequalities:
  • Misinterpreting whether an inequality is strict (< or >) or non-strict (≤ or ≥) can lead to incorrect solutions. Carefully read the inequality sign and adjust your solution accordingly.
  • Combining terms that cannot be combined in an inequality is a mistake. For example, you cannot combine "x + 3" and "2x - 5" as "3x - 2" without considering the signs and operations involved. 
  • When solving inequalities involving absolute values or quadratics, failing to identify critical points (where the expression equals zero) can lead to incorrect solutions.
  •  Some inequalities may have extraneous solutions that do not satisfy the original problem's context. Always check if your solutions make sense within the problem's context.

Conclusion - Mastering Math Inequalities

To master math inequalities, practice solving a variety of inequality problems, and double-check your work to ensure accuracy. Developing a systematic approach and paying attention to detail can help you excel in dealing with inequalities in mathematics.
One could easily practice it from Smartkeeda which have quality questions in abundance and can be accessed at https://www.smartkeeda.com/Quantitative-Aptitude/Arithmetic/Maths-Inequalities-Quiz.

Author : Amrish

Hi, this is Amrish, an Assistant Manager in one of the largest private banks with an extensive background in the banking industry. Having post-graduated with honors in Finance from a renowned university, I embarked on a rewarding career in the financial services industry. With a deep understanding of their unique needs and goals for banking exams, I would love to help aspirants through my blogs. I will keep sharing insights for cracking various exams by providing tips, strategies, mistakes one should avoid, the best resources, and much more. God Speed to all aspirants!

FAQ’s

Mathematical inequalities find numerous applications in real life across various fields and scenarios. For instance, the comparison of interest rates and returns on investments involves inequalities. In healthcare, cholesterol levels below a certain threshold are considered healthy, etc.

An equation is a mathematical statement that asserts that two expressions are equal. It is typically written in the form "expression1 = expression2." On the other hand an inequality is a mathematical statement that shows a relationship between two expressions where one is greater than, less than, or not equal to the other.

One way is start by solving the given inequality to find the range of values that satisfy it. Depending on whether it's a less than (<), less than or equal to (≤), greater than (>), or greater than or equal to (≥) inequality, your solution may be expressed in terms of a range or set of values.

Though we have discussed it detail above, but to sum up, yes, there are special rules and techniques for solving inequalities involving absolute values. These types of inequalities often require a bit of extra care because the absolute value function can create multiple cases based on whether the expression inside the absolute value is positive or negative.

Inequalities play a crucial role in various areas of mathematics, including calculus and linear algebra, as in Calculus, limits, derivatives and integrals uses the inequality is some or other way. Similarly, in linear algebra, Inequalities are used to define regions in vector spaces, and solutions to systems of linear inequalities represent feasible regions in linear programming and optimization problems.