What is a Quadratic Inequality?

S6-SA1-0235

Grade Level:

Class 10

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

Definition

What is it?

A quadratic inequality is a mathematical statement that compares a quadratic expression to a value using inequality signs like greater than (>), less than (<), greater than or equal to (>=), or less than or equal to (<=). Just like a quadratic equation has x^2, a quadratic inequality also has an x^2 term but instead of an equals sign, it has an inequality sign.

Simple Example

Quick Example

Imagine you are making a square rangoli design, and you want its area to be less than 25 square feet. If the side of the rangoli is 'x' feet, its area is x^2. So, the inequality would be x^2 < 25. This is a simple quadratic inequality where we need to find the possible lengths of the side 'x'.

Worked Example

Step-by-Step

Let's solve the quadratic inequality x^2 - 4x - 5 > 0.

1. First, treat it like an equation and find the roots of x^2 - 4x - 5 = 0.

2. Factor the quadratic expression: (x - 5)(x + 1) = 0.

3. The roots (critical points) are x = 5 and x = -1.

4. Plot these critical points on a number line. These points divide the number line into three intervals: (-infinity, -1), (-1, 5), and (5, infinity).

5. Pick a test value from each interval and substitute it into the original inequality x^2 - 4x - 5 > 0.
- For (-infinity, -1), let's pick x = -2: (-2)^2 - 4(-2) - 5 = 4 + 8 - 5 = 7. Since 7 > 0, this interval is a solution.
- For (-1, 5), let's pick x = 0: (0)^2 - 4(0) - 5 = -5. Since -5 is NOT > 0, this interval is not a solution.
- For (5, infinity), let's pick x = 6: (6)^2 - 4(6) - 5 = 36 - 24 - 5 = 7. Since 7 > 0, this interval is a solution.

6. Combine the intervals where the inequality holds true.

Answer: x < -1 or x > 5.

Why It Matters

Quadratic inequalities help engineers design safe bridges or plan satellite trajectories by ensuring certain conditions are met, like a force being within a specific range. Doctors might use them to model drug dosages, ensuring the amount of medicine in the body stays above a minimum level but below a maximum. AI/ML algorithms also use these concepts for optimization problems, making them super useful for careers in technology and science!

Common Mistakes

MISTAKE: Dividing by a negative number without reversing the inequality sign. For example, if -2x > 6, students might write x > -3. | CORRECTION: Always reverse the inequality sign when multiplying or dividing both sides by a negative number. So, -2x > 6 becomes x < -3.

MISTAKE: Only finding the roots and assuming the solution is between them, or outside them, without testing intervals. For example, for x^2 - 9 < 0, roots are -3 and 3, and students might guess x < -3 or x > 3. | CORRECTION: After finding the roots, always test a value from each interval created on the number line to see if it satisfies the original inequality.

MISTAKE: Incorrectly factoring or solving the related quadratic equation. For example, solving x^2 - 5x + 6 = 0 and getting roots as 1 and 6 instead of 2 and 3. | CORRECTION: Double-check your factoring or use the quadratic formula carefully to find the correct roots (critical points) of the associated equation.

Practice Questions

Try It Yourself

QUESTION: Solve x^2 - 9 <= 0 | ANSWER: -3 <= x <= 3

QUESTION: Find the values of x for which x^2 + 2x - 8 > 0 | ANSWER: x < -4 or x > 2

QUESTION: A mobile phone company wants its profit P (in lakhs of rupees) to be at least 15 lakhs. The profit is given by the expression P = x^2 - 10x + 40, where x is the number of phones sold (in thousands). Find the range of x for which the profit is at least 15 lakhs. | ANSWER: x <= 5 or x >= 15 (assuming x is in thousands, so x=5 means 5000 phones and x=15 means 15000 phones)

MCQ

Quick Quiz

Which of the following is NOT a quadratic inequality?

x^2 + 3x - 10 > 0

2x^2 <= 50

x^3 - 4x + 1 < 0

(x - 1)(x + 2) >= 0

The Correct Answer Is:

A quadratic inequality must have the highest power of 'x' as 2. Option C has x^3, making it a cubic inequality, not quadratic. Options A, B, and D all have x^2 as the highest power.

Real World Connection

In the Real World

Imagine you're an engineer designing a ramp for a wheelchair. The angle of the ramp depends on its length 'x'. You might have an inequality like x^2 - 10x + 24 <= 0 to ensure the ramp's slope is not too steep for safety. Or, an economist might use quadratic inequalities to determine the range of production levels for a factory to keep costs below a certain budget.

Key Vocabulary

Key Terms

QUADRATIC EXPRESSION: An expression with the highest power of the variable as 2, like ax^2 + bx + c | INEQUALITY: A mathematical statement comparing two expressions using symbols like >, <, >=, <= | CRITICAL POINTS: The roots of the related quadratic equation, which divide the number line into intervals | INTERVAL: A set of numbers between two given numbers | SOLUTION SET: The range of values for the variable that satisfy the inequality

What's Next

What to Learn Next

Great job understanding quadratic inequalities! Next, you should explore 'Systems of Quadratic Inequalities'. This will teach you how to solve problems where you have more than one quadratic inequality at the same time, which is very common in real-world applications.