Quadratic inequalities are inequalities that involve quadratic expressions. These inequalities can be either greater than, greater than or equal to, less than, or less than or equal to a quadratic expression.
To solve a quadratic inequality, we usually follow these steps:
1. Rewrite the quadratic inequality in standard form, with zero on one side of the inequality symbol.
2. Factor the quadratic expression, if possible.
3. Determine the critical points by setting each factor equal to zero and solving for the variable.
4. Use a number line to plot the critical points and choose test points from the intervals between the critical points.
5. Substitute the test points into the original inequality to determine which intervals satisfy the inequality.
6. Write the solution using interval notation or inequality notation.
For example, let's solve the quadratic inequality: x^2 - 5x + 6 > 0.
1. Rewrite the inequality: x^2 - 5x + 6 > 0.
2. Factor the quadratic expression: (x - 2)(x - 3) > 0.
3. Determine the critical points: x = 2 and x = 3.
4. Use a number line and plot the critical points:
---o---o---
5. Choose test points:
- For the interval (-∞, 2), let's choose x = 0.
- For the interval (2, 3), let's choose x = 2.5.
- For the interval (3, +∞), let's choose x = 4.
6. Substitute the test points:
- For x = 0, (0 - 2)(0 - 3) > 0 -> 2 * 3 > 0 -> 6 > 0, which is true.
- For x = 2.5, (2.5 - 2)(2.5 - 3) > 0 -> 0.5 * (-0.5) > 0 -> -0.25 > 0, which is false.
- For x = 4, (4 - 2)(4 - 3) > 0 -> 2 * 1 > 0 -> 2 > 0, which is true.
7. Write the solution using interval notation: (-∞, 2) U (3, +∞).
quadratic inequalities
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