What is Solving Quadratic Inequalities?

S3-SA1-0385

Grade Level:

Class 7

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

Solving quadratic inequalities means finding the range of values for a variable (like 'x') that make a quadratic expression true when compared using symbols like 'greater than' (>) or 'less than' (<). A quadratic expression is one where the highest power of the variable is 2, like x^2 + 2x - 3.

Simple Example

Quick Example

Imagine you're designing a small garden patch. The length is 'x' meters and the width is 'x - 2' meters. If you want the area of the garden to be less than 8 square meters, you'd write this as x(x - 2) < 8. Solving this inequality helps you find the possible lengths 'x' for your garden.

Worked Example

Step-by-Step

Let's solve the inequality x^2 - 4x + 3 < 0.

Step 1: Convert the inequality into an equation and find its roots. So, x^2 - 4x + 3 = 0.
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Step 2: Factor the quadratic equation. (x - 1)(x - 3) = 0.
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Step 3: Find the roots. x - 1 = 0 gives x = 1, and x - 3 = 0 gives x = 3. These are called critical points.
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Step 4: Draw a number line and mark these critical points (1 and 3). These points divide the number line into three intervals: (-infinity, 1), (1, 3), and (3, +infinity).
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Step 5: Pick a test value from each interval and substitute it into the original inequality x^2 - 4x + 3 < 0.
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Step 6: For interval (-infinity, 1), let's pick x = 0. (0)^2 - 4(0) + 3 = 3. Is 3 < 0? No, it's false.
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Step 7: For interval (1, 3), let's pick x = 2. (2)^2 - 4(2) + 3 = 4 - 8 + 3 = -1. Is -1 < 0? Yes, it's true.
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Step 8: For interval (3, +infinity), let's pick x = 4. (4)^2 - 4(4) + 3 = 16 - 16 + 3 = 3. Is 3 < 0? No, it's false.
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Answer: The interval where the inequality is true is (1, 3). So, 1 < x < 3.

Why It Matters

Understanding quadratic inequalities is super useful in many advanced fields. For example, Data Scientists use them to define ranges for data analysis, and Engineers use them to design structures, ensuring safety limits are met. Even AI/ML models use these concepts to optimize their performance, helping us build smarter apps!

Common Mistakes

MISTAKE: Treating inequalities like equations and only finding the roots, without checking intervals. | CORRECTION: After finding the roots, always test values in the intervals created by these roots on a number line to determine where the inequality holds true.

MISTAKE: Forgetting to reverse the inequality sign when multiplying or dividing by a negative number. | CORRECTION: Remember that if you multiply or divide both sides of an inequality by a negative number, you MUST flip the inequality sign (e.g., < becomes >, and > becomes <).

MISTAKE: Incorrectly assuming the solution is always 'between' the roots or 'outside' the roots without proper testing. | CORRECTION: The direction of the parabola (upwards if x^2 coefficient is positive, downwards if negative) and the inequality sign (>, <, >=, <=) determine the solution. Always test points.

Practice Questions

Try It Yourself

QUESTION: Solve x^2 - 5x + 6 > 0 | ANSWER: x < 2 or x > 3

QUESTION: Solve 2x^2 + 7x + 3 <= 0 | ANSWER: -3 <= x <= -1/2

QUESTION: A mobile game developer wants to set a price 'p' for a new in-game item. The profit is given by P(p) = -p^2 + 10p - 21. For what range of prices 'p' will the profit be positive (P(p) > 0)? | ANSWER: 3 < p < 7

MCQ

Quick Quiz

Which of the following is the correct solution for x^2 - 9 < 0?

x < -3 or x > 3

-3 < x < 3

x = 3

x = -3

The Correct Answer Is:

For x^2 - 9 < 0, the roots of x^2 - 9 = 0 are x = 3 and x = -3. Testing a value like x=0 (between -3 and 3) gives 0^2 - 9 = -9, which is less than 0. So, -3 < x < 3 is the correct solution.

Real World Connection

In the Real World

Imagine you're managing inventory for a popular online clothing store like Myntra. You might use quadratic inequalities to figure out the optimal price range for a new line of kurtas to maximize profit, while also ensuring you sell enough stock without having too much leftover. Or, an engineer designing a flyover might use them to calculate safe load limits, ensuring the bridge can handle traffic without stress exceeding a certain limit.

Key Vocabulary

Key Terms

QUADRATIC EXPRESSION: An expression where the highest power of the variable is 2, like ax^2 + bx + c. | INEQUALITY: A mathematical statement comparing two expressions using symbols like <, >, <=, or >=. | ROOTS (or ZEROS): The values of the variable that make a quadratic equation equal to zero. | CRITICAL POINTS: The roots of the quadratic equation, which divide the number line into intervals for testing.

What's Next

What to Learn Next

Great job understanding quadratic inequalities! Next, you can explore 'Systems of Linear and Quadratic Inequalities'. This will teach you how to solve problems involving multiple conditions at once, which is super helpful for more complex real-world scenarios.