How to Solve Inequalities in Math: Step-by-Step Guide

The concept of inequalities is essential in mathematics and helps in solving real-world and exam-level problems efficiently. Understanding inequalities allows students to analyze comparisons, optimize situations, and solve various algebraic challenges easily.

Understanding Inequalities

An inequality in mathematics is a statement that compares two values or expressions and shows that they are not necessarily equal. Instead, one can be greater than, less than, or sometimes equal to the other. This concept is widely used in Linear Inequalities, Algebraic Expressions, and Graphical Representation. The basic symbols used in inequalities include > (greater than), < (less than), ≥ (greater than or equal to), ≤ (less than or equal to), and ≠ (not equal to). You encounter inequalities in daily life, such as age restrictions, budget limits, and speed limits.

Inequality Symbols and What They Mean

Here’s a helpful table to understand inequality symbols more clearly:

Inequality Symbols Table

Symbol	Meaning	Example
>	Greater than	x > 7
<	Less than	a < 4
≥	Greater than or equal to	y ≥ 2
≤	Less than or equal to	b ≤ 15
≠	Not equal to	m ≠ n

These symbols make it easy to state rules or boundaries in math problems and everyday events.

Types of Inequalities

There are several types of inequalities that appear commonly in maths:

• Strict inequalities: use < or > (x < 5, x > 2)
• Non-strict inequalities: use ≤ or ≥ (y ≥ 4, a ≤ 8)
• Linear inequalities: expressions to the first power only (2x + 3 > 5)
• Quadratic inequalities: involve x² or higher-degree terms (x² - 4 < 0)
• Absolute value and rational inequalities: involve |x| or fractions

Rules for Working with Inequalities

There are some key rules to follow when solving inequalities:

• You can add or subtract the same value on both sides with no effect on the inequality direction.
• You can multiply or divide both sides by a positive number without changing the sign.
• Important! If you multiply or divide by a negative number, reverse the inequality sign.
• If you take reciprocals (when both sides are positive), the sign will flip.

Care is needed to avoid common mistakes, especially with multiplying or dividing by negatives.

How to Write Inequalities in Words

It’s important to convert mathematical statements with inequality symbols into words, and vice versa. Here are some examples:

Inequality Notation	In Words
x < 12	x is less than 12
y ≥ 5	y is greater than or equal to 5
p ≠ q	p is not equal to q

Practising these conversions helps you understand how inequalities describe real-world situations.

Worked Example – Solving a Problem

Let’s solve the inequality: \( 2x + 3 > 7 \)

1. Subtract 3 from both sides:

2. Simplify:

3. Divide both sides by 2:

Final Answer: The solution is all values of x greater than 2.

Graphical Representation of Inequalities

Inequalities are often shown on a number line. For example, to graph x < 4:

1. Draw a number line and mark the point 4.
2. Use an open circle at 4 (since it’s not ≤).
3. Shade the line to the left of 4 to show all values less than 4.

If the inequality had been x ≤ 4, you’d use a filled circle. This visual method helps quickly identify the solution set, and you can learn more about this on the linear inequalities in two variables page.

Practice Problems

Try these problems to check your understanding of inequalities:

• Solve: \( 3x - 5 \leq 10 \)
• Express in words: y > -1
• Graph x ≥ 6 on a number line.
• Solve and represent: \( x^2 - 4 < 0 \)

Common Mistakes to Avoid

• Forgetting to flip the inequality sign when multiplying or dividing by a negative number.
• Using closed circles for > or < on a number line instead of open circles.
• Not checking excluded values when dealing with rational inequalities.

Real-World Applications

The concept of inequalities appears in finance (setting spending limits), science (temperature ranges), and engineering (material strength limits). At Vedantu, teachers often show students how inequalities are used in exam settings and everyday decision-making. Seeing maths in action makes it easier to learn and remember!

We explored the idea of inequalities, how to apply them, solve related problems, and understand their real-life relevance. Practice with various questions, review rules, and try sample problems on Vedantu to build confidence in handling inequalities.

Explore More on Inequalities

For deeper understanding and more examples, check out these useful pages:
Solving Inequalities | Linear Inequalities – Class 11 | Linear Inequalities in Two Variables | Solving Rational Inequalities | Rules of Inequality | Linear Equations in Two Variables