how to graph quadeatic inequalities on a number line

To graph quadratic inequalities on a number line, follow these steps:

### Steps:
1. **Solve the corresponding quadratic equation**: First, solve the quadratic inequality as if it were an equation (e.g., \( ax^2 + bx + c = 0 \)) to find its roots. These will be the critical points where the expression equals zero.

2. **Plot the roots on the number line**: Place the solutions (roots) of the quadratic equation on the number line. These will divide the number line into intervals.

3. **Test the intervals**: Pick a test point from each interval and substitute it into the original inequality. Determine whether the inequality holds for each interval. 
    - If the result satisfies the inequality, that interval will be part of the solution.
    - If not, it won’t be included.

4. **Check inequality type**: 
    - For **strict inequalities** (e.g., \( < \) or \( > \)), the critical points (roots) themselves are not part of the solution, so they will be marked with **open circles**.
    - For **non-strict inequalities** (e.g., \( \leq \) or \( \geq \)), the critical points are included in the solution and are marked with **closed circles**.

5. **Shade the appropriate regions**: Based on the test results, shade the intervals where the inequality holds. This shading represents the solution to the inequality.

### Example:
Graph the inequality \( x^2 - 4x + 3 \leq 0 \).

1. **Solve the quadratic equation**: 
   \( x^2 - 4x + 3 = 0 \).
   Factor: \( (x - 1)(x - 3) = 0 \).
   So, the roots are \( x = 1 \) and \( x = 3 \).

2. **Plot these roots on the number line**: Mark \( x = 1 \) and \( x = 3 \).

3. **Test the intervals**: The number line is divided into three intervals: \( (-\infty, 1) \), \( (1, 3) \), and \( (3, \infty) \).
   - Test \( x = 0 \) in \( (-\infty, 1) \): \( 0^2 - 4(0) + 3 = 3 \), which is **not less than or equal to 0**.
   - Test \( x = 2 \) in \( (1, 3) \): \( 2^2 - 4(2) + 3 = -1 \), which **is less than or equal to 0**.
   - Test \( x = 4 \) in \( (3, \infty) \): \( 4^2 - 4(4) + 3 = 3 \), which is **not less than or equal to 0**.

4. **Include or exclude the points**: Since it's \( \leq \), include \( x = 1 \) and \( x = 3 \), so they are marked with closed circles.

5. **Shade the solution region**: The solution is the interval \( [1, 3] \), so you would shade that portion of the number line.

Would you like further details or have any questions?

### 5 related questions:
1. How do you graph a quadratic inequality that has no real roots?
2. What changes when graphing strict inequalities (e.g., \( > \) or \( < \))?
3. How do you graph inequalities involving quadratic expressions with negative leading coefficients?
4. How do quadratic inequalities relate to their corresponding parabolas?
5. What is the significance of the direction of the inequality symbol in quadratic inequalities?

### Tip:
Always remember to check your test points from each interval to ensure accuracy when shading your solution region.

how to graph quadratic inequalities on a number line

Learn how to graph quadratic inequalities on a number line with this step-by-step guide. This explanation covers solving quadratic equations, plotting roots, testing intervals, and shading solution regions, making it ideal for students in Grades 8-10.

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