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How are quadratic inequalities different from linear inequalities?

Published in Quadratic Inequalities 3 mins read

Quadratic inequalities differ from linear inequalities primarily in the degree of the variable involved. According to our reference material, a linear inequality involves only variables to the first power, while a quadratic inequality involves variables to the second power, and possibly also to the first power. This difference in structure leads to different methods for solving them.

Key Differences Summarized

Here's a table summarizing the key distinctions:

Feature Linear Inequality Quadratic Inequality
Variable's Power Variables are raised to the first power only. Variables are raised to the second power (and possibly first).
General Form ax + b < 0 (or >, ≤, ≥) ax2 + bx + c < 0 (or >, ≤, ≥)
Solution Method Isolating the variable. Factoring, using the quadratic formula, testing intervals.
Solution Set Usually a single interval or ray. Often two separate intervals or a single interval.

Detailed Explanation

Let's elaborate on these differences:

  • Variable's Power: The most fundamental difference is the highest power of the variable. Linear inequalities deal with terms like x, while quadratic inequalities involve terms like x2.

  • General Form: A linear inequality can be generally represented as ax + b < 0, where a and b are constants. A quadratic inequality has the general form ax2 + bx + c < 0, where a, b, and c are constants, and a cannot be zero. The "<" symbol can be replaced with ">", "≤", or "≥".

  • Solution Method: Linear inequalities are typically solved by isolating the variable on one side of the inequality. For example, to solve 2x + 3 < 7, you would subtract 3 from both sides and then divide by 2. Quadratic inequalities require a more complex approach.

    • Steps to Solve Quadratic Inequalities:
      1. Rewrite the inequality so that one side is zero.
      2. Find the roots of the corresponding quadratic equation (e.g., ax2 + bx + c = 0). This can be done by factoring, completing the square, or using the quadratic formula. These roots are the critical values.
      3. Create a number line and mark the critical values.
      4. Choose test values from the intervals created by the critical values. Substitute these test values into the original inequality to determine if the inequality holds true in that interval.
      5. Write the solution set based on the intervals where the inequality is true.

  • Solution Set: The solution to a linear inequality is typically a single interval or ray on the number line (e.g., x < 2). The solution to a quadratic inequality can be a single interval, two separate intervals, or even no solution at all.

    • Example: The solution to x2 - 3x + 2 < 0 is 1 < x < 2, a single interval. The solution to x2 - 3x + 2 > 0 is x < 1 or x > 2, representing two separate intervals.

Examples

Let's illustrate with examples:

  • Linear Inequality: 3x - 5 > 4

    • Solution: x > 3

  • Quadratic Inequality: x2 - 4x + 3 ≤ 0

    • Factoring: (x - 1)(x - 3) ≤ 0
    • Critical values: x = 1, x = 3
    • Testing intervals:
      • x < 1: False (e.g., x = 0: (0-1)(0-3) = 3 > 0)
      • 1 < x < 3: True (e.g., x = 2: (2-1)(2-3) = -1 < 0)
      • x > 3: False (e.g., x = 4: (4-1)(4-3) = 3 > 0)
    • Solution: 1 ≤ x ≤ 3

In essence, the presence of the squared term in quadratic inequalities makes them fundamentally different from linear inequalities, necessitating different solving techniques and resulting in potentially more complex solution sets.

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