Solve Inequalities Using Multiplication and Division: Step-by-Step Guide

Solving Inequalities By Using Different Operations Using Multiplication And Division is a crucial concept in algebra that students encounter in school maths and competitive exams like JEE or Olympiad. Understanding how to manipulate inequalities using multiplication or division is essential for solving a wide range of problems, both in academics and real-life decision-making.

Solving Inequalities Using Multiplication and Division: Core Concept

An inequality is a mathematical statement that compares two values or expressions using symbols like <, >, ≤, or ≥. When solving inequalities, you aim to find all possible values of the variable that make the inequality true. Just like equations, you can use multiplication and division to isolate the variable, but with special rules to watch out for—especially when dealing with negative numbers.

These skills are foundational for topics like linear equations in one variable, general inequalities, and operations on rational numbers. Getting comfortable with these operations now helps in advanced chapters later.

Basic Properties: Inequality Signs and When to Flip Them

• < means "less than"
• > means "greater than"
• ≤ means "less than or equal to"
• ≥ means "greater than or equal to"

When you multiply or divide both sides of an inequality by a positive number, the inequality sign does NOT change. But if you multiply or divide both sides by a negative number, you MUST reverse (flip) the sign. This is called the sign reversal rule.

Key Rules for Multiplication and Division in Inequalities

• Multiplying/Dividing by a positive number: Keep the inequality sign the same.
• Multiplying/Dividing by a negative number: Reverse the sign (< becomes >, > becomes <, ≤ becomes ≥, ≥ becomes ≤).

These are called the Multiplication Property of Inequality and Division Property of Inequality. For more on the underlying reason, visit Multiplication and Division of Integers – Rules.

Worked Examples: Step-by-Step Solutions

Example 1: Multiplying by a Positive Number

Solve for $x$ : $\frac{x}{4}>2$

1. Multiply both sides by 4 (positive number):
2. $x>8$
3. All values greater than 8 are solutions.

Example 2: Dividing by a Negative Number (Sign Flip)

Solve for $x$ : $-2x\le 6$

1. Divide both sides by -2 (negative number):
2. Don't forget to flip the sign!
3. $x\ge -3$

Example 3: Multiplying by a Negative Number (Sign Flip)

Solve for $x$ : $5<-x$

1. Multiply both sides by -1 (negative number):
2. Flip the sign:
3. $-5>x$ ⇒ $x<-5$

Formula and Property Summary

• If $a<b$ and $c>0$, then $a\times c<b\times c$
• If $a<b$ and $c<0$, then $a\times c>b\times c$
• If $a<b$ and $c>0$, then $\frac{a}{c}<\frac{b}{c}$
• If $a<b$ and $c<0$, then $\frac{a}{c}>\frac{b}{c}$

Practice Problems

• Solve for $x$: $x/5\ge 2$
• Solve for $x$: $-3x<12$
• Solve for $x$: $2x/7>-4$
• Solve for $x$: $x/(-2)\le -5$
• Solve for $x$: $-x\ge 9$
• Solve for $x$: $7x>21$
• Solve for $x$: $-4x\le 20$
• Solve for $x$: $x/3<0$
• Solve for $x$: $10\le -2x$
• Solve for $x$: $x/(-5)\ge 2$

You can find more worksheets on solving inequalities and multiplication/division properties on Vedantu.

Common Mistakes to Avoid

• Forgetting to flip the sign when multiplying or dividing both sides by a negative number.
• Multiplying or dividing by zero (never do this; it's undefined or not allowed in inequalities).
• Treating inequalities exactly like equations in all operations—remember, only multiply/divide by positive numbers without flipping the sign.
• Confusing “less than” with “greater than” after sign reversal—draw a number line if unsure.

Real-World Applications

These methods are not just for classwork. For example, if a worker is paid more than ₹200 a day, and you know for 5 days the payment is above ₹1000: $5x>1000$. Dividing both sides by 5 gives $x>200$. Also, when budgeting, comparing rates, or checking speed limits, you often see such inequalities in real life.

Page Summary

In summary, Solving Inequalities By Using Different Operations Using Multiplication And Division teaches you to handle inequalities confidently by applying multiplication and division rules with care—especially for negatives. Mastering these rules is a key step towards strong mathematical foundations and success in school and beyond. For more personalized help and extensive practice, explore topics and live classes at Vedantu.