\[ y < ax^2 + bx + c \quad \text{or} \quad y \leq ax^2 + bx + c \] \[ y > ax^2 + bx + c \quad \text{or} \quad y \geq ax^2 + bx + c \]

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

Graph: \( y \leq x^2 - 4x + 3 \)

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)

Solve: \( x^2 - 5x + 6 < 0 \)

Solve: \( x^2 + 3x \geq 10 \)

\[ y < P(x) \quad \text{or} \quad y \leq P(x) \quad \text{or} \quad y > P(x) \quad \text{or} \quad y \geq P(x) \]

where \( P(x) \) is a polynomial of degree \( \geq 3 \)

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 \( \geq \)

• 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

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

Solve: \( x^3 - 4x > 0 \)

Solve: \( x^4 - 5x^2 + 4 \leq 0 \)

• 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

• 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

✗ Forgetting to test all intervals

✗ Using wrong bracket notation ([ ] vs ( ))

✗ Incorrectly determining sign in each interval

✗ Not considering multiplicity when analyzing sign changes

Interval Notation:

• Use ( ) for < or > (open)

• Use [ ] for ≤ or ≥ (closed)

• Use ∪ to combine multiple intervals