What is Solving Polynomial Inequalities?

S6-SA1-0347

Grade Level:

Class 10

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

Definition

What is it?

Solving Polynomial Inequalities means finding the range of 'x' values for which a polynomial expression is either greater than, less than, greater than or equal to, or less than or equal to zero. It's like finding where a polynomial graph is above or below the x-axis.

Simple Example

Quick Example

Imagine you are selling samosas, and your profit P depends on the number of samosas 'x' you sell, given by the polynomial P = x^2 - 10x + 24. If you want to find out for what number of samosas you make a profit (P > 0), you would solve the polynomial inequality x^2 - 10x + 24 > 0. This helps you know how many samosas you need to sell to be in profit.

Worked Example

Step-by-Step

Solve the inequality: x^2 - 5x + 6 > 0

1. Find the roots of the corresponding polynomial equation: x^2 - 5x + 6 = 0.
Factoring the quadratic equation, we get (x - 2)(x - 3) = 0.
The roots are x = 2 and x = 3.

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2. Mark these roots on a number line. These roots divide the number line into three intervals: (-infinity, 2), (2, 3), and (3, infinity).

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3. Choose a test value from each interval and substitute it into the original inequality x^2 - 5x + 6 > 0.
- For interval (-infinity, 2), let's pick x = 0: (0)^2 - 5(0) + 6 = 6. Since 6 > 0, this interval satisfies the inequality.

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4. - For interval (2, 3), let's pick x = 2.5: (2.5)^2 - 5(2.5) + 6 = 6.25 - 12.5 + 6 = -0.25. Since -0.25 is NOT > 0, this interval does not satisfy the inequality.

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5. - For interval (3, infinity), let's pick x = 4: (4)^2 - 5(4) + 6 = 16 - 20 + 6 = 2. Since 2 > 0, this interval satisfies the inequality.

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6. The intervals where the inequality is satisfied are (-infinity, 2) and (3, infinity).

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ANSWER: The solution is x < 2 or x > 3, which can also be written as (-infinity, 2) U (3, infinity).

Why It Matters

Solving polynomial inequalities is super important for engineers designing structures, scientists predicting chemical reactions, and even in AI/ML for optimizing algorithms. For example, in physics, it helps understand when a projectile's height is above a certain level, guiding careers in space technology or aerospace engineering.

Common Mistakes

MISTAKE: Multiplying or dividing by a variable term without considering its sign. | CORRECTION: When multiplying or dividing both sides of an inequality by a negative number or a variable term whose sign is unknown, you must reverse the inequality sign.

MISTAKE: Incorrectly combining intervals or forgetting to check boundary points. | CORRECTION: Always test a value from each interval created by the roots on the number line. For inequalities with 'greater than or equal to' or 'less than or equal to' signs, include the roots in your solution interval.

MISTAKE: Only finding the roots and assuming the solution is between them. | CORRECTION: The roots only tell you where the polynomial equals zero. You need to test values in the intervals between and outside the roots to see where the inequality holds true.

Practice Questions

Try It Yourself

QUESTION: Solve: x^2 - 4 < 0 | ANSWER: -2 < x < 2

QUESTION: Solve: (x - 1)(x + 2)(x - 3) >= 0 | ANSWER: [-2, 1] U [3, infinity)

QUESTION: A mobile app's daily user growth 'G' depends on the number of marketing ads 'n' as G = n^2 - 8n + 15. For what range of 'n' will the user growth be positive (G > 0)? | ANSWER: n < 3 or n > 5

MCQ

Quick Quiz

Which of the following values of x satisfies the inequality x^2 - 7x + 10 < 0?

x = 1

x = 3

x = 6

x = 0

The Correct Answer Is:

The roots of x^2 - 7x + 10 = 0 are x = 2 and x = 5. For x^2 - 7x + 10 < 0, the solution is 2 < x < 5. Only x = 3 falls within this range.

Real World Connection

In the Real World

Imagine a drone delivering packages for a company like Zomato or Swiggy. Engineers use polynomial inequalities to determine the safe flight altitude range for the drone, ensuring it stays above obstacles but below restricted airspace. This ensures smooth and safe deliveries across Indian cities.

Key Vocabulary

Key Terms

POLYNOMIAL: An expression with one or more terms, each having variables raised to non-negative integer powers | INEQUALITY: A mathematical statement showing that two expressions are not equal, using symbols like <, >, <=, or >= | ROOTS: The values of the variable that make a polynomial equal to zero | INTERVAL: A set of real numbers between two given numbers | NUMBER LINE: A visual representation of real numbers as points on a line

What's Next

What to Learn Next

Next, you can explore 'Rational Inequalities' and 'Absolute Value Inequalities'. These concepts build on your understanding of polynomial inequalities by adding fractions and absolute values, making them useful for even more complex real-world problems.