How To Solve Quadratic Inequalities With Snake Method

Solving quadratic inequalities can be challenging, but the Snake Method offers a visual, intuitive approach that simplifies the process. This method helps you determine which sections of a parabola satisfy the inequality by focusing on the sign changes along the number line. In this guide, we’ll break down how to apply the Snake Method step-by-step.

Why Use the Snake Method?

• Visual Clarity: Makes it easier to see where the inequality holds true.
• Quick Process: Saves time compared to traditional algebraic methods.
• Ideal for Exams: Great for standardized tests where speed and accuracy matter.

Tools You’ll Need

• Pen and paper
• Basic knowledge of factoring quadratics
• Optional: Graphing calculator for verification

Steps to Solve Quadratic Inequalities with the Snake Method

1. Write the Quadratic Inequality in Standard Form

Ensure your inequality is in the form:

ax2+bx+c ≶ 0ax^2 + bx + c \, \lessgtr \, 0

Example:

x2−5x+6>0x^2 – 5x + 6 > 0

2. Factor the Quadratic Expression

Factor the quadratic (if possible):

(x−2)(x−3)>0(x – 2)(x – 3) > 0

Identify the critical points (roots): x=2x = 2 and x=3x = 3.

3. Plot the Critical Points on a Number Line

• Draw a number line and mark the roots x=2x = 2 and x=3x = 3.
• These points divide the number line into three intervals:
  1. (−∞,2)(-\infty, 2)
  2. (2,3)(2, 3)
  3. (3,∞)(3, \infty)

4. Apply the Snake Method (Sign Analysis)

• Draw a “snake” curve that alternates between positive and negative signs as it passes through each interval.
• The snake starts with a positive sign to the right of the largest root if the leading coefficient aa is positive.
• Alternate signs as you move left across the critical points.

Example:

• For (3,∞)(3, \infty): Positive (+)
• Between 22 and 33: Negative (−)
• For (−∞,2)(-\infty, 2): Positive (+)

5. Identify the Solution Based on the Inequality

• For >0> 0, choose the positive intervals: (−∞,2)(-\infty, 2) and (3,∞)(3, \infty)
• For <0< 0, choose the negative interval: (2,3)(2, 3)
• Exclude or include the critical points based on whether the inequality is strict (>> or <<) or inclusive (≥\geq or ≤\leq).

Final Answer:

x<2orx>3x < 2 \quad \text{or} \quad x > 3

Tips for Using the Snake Method Effectively

• Always factor first: Factoring simplifies the process and makes sign analysis easier.
• Watch the leading coefficient: It determines the sign pattern’s starting point.
• Sketch the parabola (optional): This can help visualize the inequality if needed.

Troubleshooting Common Issues

• Can’t Factor the Quadratic?
  
  • Use the quadratic formula to find roots and apply the Snake Method to those values.
• Incorrect Sign Pattern?
  
  • Double-check the sign of the leading coefficient to ensure you start with the correct sign.
• Confused About Including Critical Points?
  
  • If the inequality is ≥\geq or ≤\leq, include the roots in the solution (closed circles on the number line).

Also Read: How To Skip Bokura Cut Scenes

Conclusion

The Snake Method is a powerful, quick way to solve quadratic inequalities by focusing on sign changes across intervals. Whether you’re preparing for an exam or just want a clearer understanding of quadratic behavior, this method offers both speed and accuracy.