What are the key steps to solve a quadratic inequality?

The key steps to solving a quadratic inequality involve finding the critical values, testing intervals, and expressing the solution set.

Here's a breakdown of the steps:

1. Rewrite the inequality into standard form: Ensure the inequality is in the form ax² + bx + c > 0, ax² + bx + c < 0, ax² + bx + c ≥ 0, or ax² + bx + c ≤ 0. This means one side should be zero.

2. Solve the related quadratic equation: Replace the inequality sign with an equals sign and solve the quadratic equation ax² + bx + c = 0. You can solve this by:

   • Factoring: If the quadratic expression is easily factorable.
   • Using the quadratic formula: x = (-b ± √(b² - 4ac)) / (2a)
   • Completing the square: If factoring is difficult.

3. Identify critical values: The solutions from the quadratic equation are your critical values (also known as boundary points or roots). These values divide the number line into intervals.

4. Create a number line and test intervals: Draw a number line and mark the critical values on it. These points divide the number line into intervals. Choose a test value within each interval and substitute it into the original inequality.

5. Determine if the test value satisfies the inequality: If the test value makes the inequality true, then all values in that interval are solutions. If it makes the inequality false, then no values in that interval are solutions.

6. Express the solution:

   • Interval Notation: Use interval notation to represent the solution set. Use parentheses "(" and ")" for intervals where the critical values are not included (strict inequalities > or <). Use brackets "[" and "]" for intervals where the critical values are included (inequalities with equality ≥ or ≤). Use "∪" to join multiple intervals.
   • Set Notation: Alternatively, you can use set notation to express the solution.
   • Graph on a Number Line: Represent the solution graphically on the number line. Use open circles (o) for endpoints not included and closed circles (●) for endpoints included.

Example:

Solve the inequality x² - 3x - 4 > 0

1. Standard form: Already in standard form.

2. Solve the equation: x² - 3x - 4 = 0. Factoring gives (x - 4)(x + 1) = 0. So, x = 4 and x = -1.

3. Critical values: -1 and 4.

4. Number line and test intervals:

   • Interval 1: x < -1. Test value: x = -2. (-2)² - 3(-2) - 4 = 4 + 6 - 4 = 6 > 0. True.
   • Interval 2: -1 < x < 4. Test value: x = 0. (0)² - 3(0) - 4 = -4 > 0. False.
   • Interval 3: x > 4. Test value: x = 5. (5)² - 3(5) - 4 = 25 - 15 - 4 = 6 > 0. True.

5. Determine solution: Intervals 1 and 3 satisfy the inequality.

6. Express the solution:

   • Interval notation: (-∞, -1) ∪ (4, ∞)
   • Set notation: {x | x < -1 or x > 4}