What is a Linear Inequality in One Variable?

S6-SA1-0109

Grade Level:

Class 10

AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine

Definition

What is it?

A linear inequality in one variable is a mathematical statement that compares two expressions using an inequality symbol (like <, >, ≤, or ≥) and contains only one variable raised to the power of one. It doesn't have an equal sign, but rather shows a range of possible values for the variable.

Simple Example

Quick Example

Imagine you have Rs. 50 to spend on chai. Each cup of chai costs Rs. 15. If 'x' is the number of chai cups you can buy, then the situation can be written as 15x ≤ 50. This is a linear inequality because it has one variable (x), it's raised to the power of one, and it uses an inequality symbol.

Worked Example

Step-by-Step

Let's solve the inequality: 3x + 7 > 22

Step 1: Subtract 7 from both sides to isolate the term with 'x'.
3x + 7 - 7 > 22 - 7
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Step 2: Simplify the equation.
3x > 15
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Step 3: Divide both sides by 3 to find 'x'.
3x / 3 > 15 / 3
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Step 4: Simplify to get the solution.
x > 5

Answer: x > 5

Why It Matters

Linear inequalities are super useful for making decisions and planning! Engineers use them to design safe structures, data scientists use them in AI to classify data, and even doctors use them to determine safe medicine dosages. Understanding them opens doors to careers in technology, science, and finance.

Common Mistakes

MISTAKE: Forgetting to flip the inequality sign when multiplying or dividing by a negative number. For example, if -2x > 10, students might write x > -5. | CORRECTION: When you multiply or divide both sides of an inequality by a negative number, you MUST reverse the inequality sign. So, -2x > 10 becomes x < -5.

MISTAKE: Treating an inequality like an equation and only finding one exact value. For example, solving x + 3 < 7 and giving 'x = 4' as the answer. | CORRECTION: Remember that an inequality represents a range of values. The solution for x + 3 < 7 is x < 4, meaning any number less than 4 is a valid solution.

MISTAKE: Not understanding what the inequality symbols mean. Forgetting if '<' includes the boundary value or not. | CORRECTION: '<' means 'less than' (boundary not included), '>' means 'greater than' (boundary not included), '≤' means 'less than or equal to' (boundary included), and '≥' means 'greater than or equal to' (boundary included).

Practice Questions

Try It Yourself

QUESTION: Solve for y: y - 8 ≤ 15 | ANSWER: y ≤ 23

QUESTION: Solve for z: 4z + 10 < 30 | ANSWER: z < 5

QUESTION: A mobile data plan costs Rs. 200 for the first 5GB and Rs. 20 for every extra GB. If you want your bill to be no more than Rs. 300, how many extra GB (let 'x' be extra GB) can you use? Write and solve the inequality. | ANSWER: 200 + 20x ≤ 300 => 20x ≤ 100 => x ≤ 5. You can use up to 5 extra GB.

MCQ

Quick Quiz

Which of the following is a linear inequality in one variable?

2x + 3y = 7

5x^2 - 1 < 0

4t - 9 ≥ 11

x + y > 10

The Correct Answer Is:

Option C (4t - 9 ≥ 11) is a linear inequality in one variable because it has only one variable ('t') raised to the power of one, and uses an inequality symbol. Option A is an equation with two variables. Option B has a variable raised to the power of two. Option D has two variables.

Real World Connection

In the Real World

Think about planning a road trip across India. If your car can travel at most 80 km/h, and you need to reach a destination 400 km away, you can use a linear inequality to figure out the minimum time needed. If 't' is time in hours, then 80t ≥ 400 helps you find t ≥ 5 hours. This helps delivery services like Zepto or Swiggy optimize routes and delivery times!

Key Vocabulary

Key Terms

VARIABLE: A letter representing an unknown value | INEQUALITY: A mathematical statement comparing two expressions using symbols like <, >, ≤, or ≥ | LINEAR: Describes an equation or inequality where the highest power of the variable is one | SOLUTION SET: The range of values that satisfy an inequality

What's Next

What to Learn Next

Great job understanding linear inequalities! Next, you can explore 'Linear Inequalities in Two Variables' to see how they apply to graphs and real-world problems involving two changing quantities. This will further build your problem-solving skills!