Nonlinear Inequalities
Complete Notes & Formulae for Twelfth Grade (Precalculus)
1. Graph Solutions to Quadratic Inequalities
Standard Forms:
\[ y < ax^2 + bx + c \quad \text{or} \quad y \leq ax^2 + bx + c \] \[ y > ax^2 + bx + c \quad \text{or} \quad y \geq ax^2 + bx + c \]
Steps to Graph:
1. Graph the parabola: Replace inequality with \( y = ax^2 + bx + c \)
• Find vertex: \( x = -\frac{b}{2a} \), then find y-coordinate
• Plot additional points and draw parabola
2. Determine boundary type:
• Solid line for \( \leq \) or \( \geq \) (boundary included)
• Dashed line for \( < \) or \( > \) (boundary not included)
3. Shade the solution region:
• \( y < \) or \( y \leq \): Shade INSIDE (below) the parabola
• \( y > \) or \( y \geq \): Shade OUTSIDE (above) the parabola
Shading Guide:
| Inequality | Boundary | Shading |
|---|---|---|
| \( y < ax^2 + bx + c \) | Dashed parabola | Inside (below) |
| \( y \leq ax^2 + bx + c \) | Solid parabola | Inside (below) |
| \( y > ax^2 + bx + c \) | Dashed parabola | Outside (above) |
| \( y \geq ax^2 + bx + c \) | Solid parabola | Outside (above) |
Example:
Graph: \( y \leq x^2 - 4x + 3 \)
Vertex: \( x = -\frac{-4}{2(1)} = 2 \), \( y = (2)^2 - 4(2) + 3 = -1 \) → (2, -1)
X-intercepts: Factor \( (x-1)(x-3) = 0 \) → x = 1, 3
Boundary: Solid parabola (opens upward)
Solution: Shade inside (below) the parabola including boundary
2. Solve Quadratic Inequalities
Algebraic Method (Sign Chart):
1. Write in standard form: Move all terms to one side, zero on the other
\( ax^2 + bx + c < 0 \) (or \( \leq \), \( > \), \( \geq \))
2. Find critical points: Solve \( ax^2 + bx + c = 0 \)
• Factor, complete the square, or use quadratic formula
• These are boundary points where expression changes sign
3. Create sign chart:
• Plot critical points on number line
• Test a value from each interval
• Mark intervals as positive (+) or negative (−)
4. Determine solution:
• For \( < 0 \): Select negative regions
• For \( > 0 \): Select positive regions
• Use [ ] for \( \leq \) or \( \geq \) (include endpoints)
• Use ( ) for \( < \) or \( > \) (exclude endpoints)
Example 1:
Solve: \( x^2 - 5x + 6 < 0 \)
Step 1: Already in standard form
Step 2: Find critical points
Factor: \( (x-2)(x-3) = 0 \)
Critical points: x = 2, x = 3
Step 3: Test intervals
Test x = 0: \( 0^2 - 5(0) + 6 = 6 > 0 \) → Positive
Test x = 2.5: \( (2.5)^2 - 5(2.5) + 6 = -0.75 < 0 \) → Negative
Test x = 4: \( 4^2 - 5(4) + 6 = 2 > 0 \) → Positive
Solution: \( (2, 3) \) or \( 2 < x < 3 \)
Example 2:
Solve: \( x^2 + 3x \geq 10 \)
Step 1: Standard form: \( x^2 + 3x - 10 \geq 0 \)
Step 2: Factor: \( (x+5)(x-2) = 0 \)
Critical points: x = -5, x = 2
Step 3: Test intervals
x < -5: Positive
-5 < x < 2: Negative
x > 2: Positive
Solution: \( (-\infty, -5] \cup [2, \infty) \) or \( x \leq -5 \) or \( x \geq 2 \)
3. Graph Solutions to Higher-Degree Inequalities
General Form:
\[ y < P(x) \quad \text{or} \quad y \leq P(x) \quad \text{or} \quad y > P(x) \quad \text{or} \quad y \geq P(x) \]
where \( P(x) \) is a polynomial of degree \( \geq 3 \)
Steps to Graph:
1. Graph the polynomial function: \( y = P(x) \)
• Find x-intercepts (zeros)
• Determine end behavior
• Plot key points and sketch curve
2. Draw boundary:
• Solid curve for \( \leq \) or \( \geq \)
• Dashed curve for \( < \) or \( > \)
3. Shade solution region:
• Below curve for \( < \) or \( \leq \)
• Above curve for \( > \) or \( \geq \)
Key Observations:
• Cubic functions (degree 3) have S-shaped curves
• Quartic functions (degree 4) have W-shaped or M-shaped curves
• End behavior depends on leading coefficient and degree
• Solution regions may be unbounded
4. Solve Higher-Degree Inequalities
General Method (Sign Chart):
1. Write in standard form: \( P(x) < 0 \) (or other inequality)
2. Find critical points: Solve \( P(x) = 0 \)
• Factor completely if possible
• Use synthetic division, rational root theorem, or graphing
3. Create sign chart:
• Plot critical points on number line
• Test value from each interval
• Mark sign of \( P(x) \) in each region
4. Select appropriate intervals:
• Choose regions where inequality is satisfied
• Write in interval notation
Example 1: Cubic Inequality
Solve: \( x^3 - 4x > 0 \)
Step 1: Factor: \( x(x^2 - 4) > 0 \)
\( x(x-2)(x+2) > 0 \)
Step 2: Critical points: x = -2, 0, 2
Step 3: Test intervals:
x = -3: \( (-3)(-5)(-1) = -15 < 0 \) → Negative
x = -1: \( (-1)(-3)(1) = 3 > 0 \) → Positive ✓
x = 1: \( (1)(-1)(3) = -3 < 0 \) → Negative
x = 3: \( (3)(1)(5) = 15 > 0 \) → Positive ✓
Solution: \( (-2, 0) \cup (2, \infty) \)
Example 2: Quartic Inequality
Solve: \( x^4 - 5x^2 + 4 \leq 0 \)
Step 1: Let \( u = x^2 \), then \( u^2 - 5u + 4 \leq 0 \)
Factor: \( (u-1)(u-4) \leq 0 \)
\( (x^2-1)(x^2-4) \leq 0 \)
Step 2: Factor completely: \( (x-1)(x+1)(x-2)(x+2) \leq 0 \)
Critical points: x = -2, -1, 1, 2
Step 3: Test intervals and determine negative regions
Solution: \( [-2, -1] \cup [1, 2] \)
5. Special Cases and Important Notes
Multiplicity of Zeros:
• Odd multiplicity: Sign changes at the zero
Example: \( (x-2)^3 \) crosses x-axis at x = 2
• Even multiplicity: Sign does NOT change at the zero
Example: \( (x-2)^2 \) touches but doesn't cross at x = 2
No Real Solutions:
• If discriminant \( b^2 - 4ac < 0 \) for quadratic:
- Parabola doesn't cross x-axis
- Solution is either all real numbers or no solution
- Depends on opening direction and inequality sign
Common Mistakes to Avoid:
✗ Forgetting to test all intervals
✗ Using wrong bracket notation ([ ] vs ( ))
✗ Incorrectly determining sign in each interval
✗ Not considering multiplicity when analyzing sign changes
6. Quick Reference Summary
Solution Process:
1. Get inequality in form \( P(x) < 0 \) (or \( \leq \), \( > \), \( \geq \))
2. Find zeros/critical points: Solve \( P(x) = 0 \)
3. Create sign chart with critical points on number line
4. Test value from each interval to determine sign
5. Select intervals where inequality is true
6. Write solution in interval notation
Interval Notation:
• Use ( ) for < or > (open)
• Use [ ] for ≤ or ≥ (closed)
• Use ∪ to combine multiple intervals
📚 Study Tips
✓ Always factor completely before creating sign chart
✓ Test points carefully - one wrong sign affects entire solution
✓ Pay attention to whether endpoints are included or excluded
✓ For graphing: solid boundary for ≤ or ≥, dashed for < or >
✓ Consider multiplicity of zeros when determining sign changes