AP Precalculus: Nonlinear Inequalities
Master quadratic and polynomial inequalities using sign charts and interval notation
📚 Understanding Nonlinear Inequalities
Nonlinear inequalities involve quadratic, polynomial, or rational expressions. Unlike linear inequalities (which create half-planes), these create more complex solution regions. The key technique is finding critical points and testing signs in each interval.
1 Understanding Quadratic Inequalities
A quadratic inequality compares a quadratic expression to zero. The solution is a set of intervals on the number line where the inequality is true.
What Solutions Mean Graphically
The roots (zeros) of the quadratic are the x-intercepts where the parabola crosses from positive to negative (or vice versa). These are the boundaries of your solution intervals!
2 Solving Quadratic Inequalities Step-by-Step
Use the sign chart method to systematically determine where the quadratic expression is positive or negative.
The 5-Step Process
- Rewrite in standard form: Get \(ax^2 + bx + c\) compared to 0
- Find the roots: Solve \(ax^2 + bx + c = 0\) using factoring, quadratic formula, etc.
- Mark critical points: Place roots on a number line, dividing it into intervals
- Test each interval: Pick a test point in each interval and evaluate the sign
- Write the solution: Use interval notation, including endpoints if \(\leq\) or \(\geq\)
Solve: \(x^2 - 5x + 6 < 0\)
Step 1: Already in standard form
Step 2: Factor: \((x-2)(x-3) = 0\) → roots: \(x = 2\) and \(x = 3\)
Step 3: Critical points divide number line into: \((-\infty, 2)\), \((2, 3)\), \((3, \infty)\)
Step 4: Test points:
Step 5: We want \(< 0\) (negative), so Solution: \((2, 3)\)
Note: Parentheses because \(<\) means endpoints NOT included
3 Impact of Parabola Direction
The sign of the leading coefficient \(a\) determines whether the parabola opens up or down, which affects where the function is positive or negative.
\(a > 0\) — Opens Upward ∪
• Positive on the outside intervals
• Negative between the roots
• \(f(x) > 0\): \((-\infty, r_1) \cup (r_2, \infty)\)
• \(f(x) < 0\): \((r_1, r_2)\)
\(a < 0\) — Opens Downward ∩
• Negative on the outside intervals
• Positive between the roots
• \(f(x) > 0\): \((r_1, r_2)\)
• \(f(x) < 0\): \((-\infty, r_1) \cup (r_2, \infty)\)
Solve: \(-x^2 + 4x - 3 \geq 0\)
Factor: \(-(x-1)(x-3) = 0\) → roots: \(x = 1\), \(x = 3\)
Since \(a < 0\): Parabola opens down → positive BETWEEN roots
Solution: \([1, 3]\) (brackets for \(\geq\))
4 Special Cases: No Real Roots or One Root
When the discriminant \(b^2 - 4ac\) is zero or negative, the quadratic has special solution sets.
No Real Roots (\(b^2 - 4ac < 0\))
Parabola doesn't cross x-axis
If \(a > 0\): Always positive
• \(f(x) > 0\): All real numbers
• \(f(x) < 0\): No solution
One Real Root (\(b^2 - 4ac = 0\))
Parabola touches x-axis at vertex
If \(a > 0\): Only equals 0 at vertex
• \(f(x) > 0\): All except vertex
• \(f(x) \geq 0\): All real numbers
Solve: \(x^2 + 2x + 5 > 0\)
Discriminant: \(4 - 20 = -16 < 0\) → no real roots
Since \(a = 1 > 0\): Parabola opens up and stays above x-axis
Solution: \((-\infty, \infty)\) or "all real numbers"
5 Higher-Degree Polynomial Inequalities
For polynomials of degree 3 or higher, the same sign chart method applies. Find all real roots and test intervals.
Steps for Higher-Degree
- Write inequality with polynomial compared to 0
- Factor completely to find all real roots
- Place all roots on number line (creating \(n+1\) intervals for \(n\) distinct roots)
- Test one point in each interval
- Write solution using interval notation
Solve: \(x^3 - 4x \leq 0\)
Factor: \(x(x^2 - 4) = x(x-2)(x+2) = 0\)
Roots: \(x = -2, 0, 2\)
Intervals: \((-\infty, -2)\), \((-2, 0)\), \((0, 2)\), \((2, \infty)\)
Test:
• \(x = -3\): \((-3)(-5)(-1) = -15 < 0\) ✓
• \(x = -1\): \((-1)(-3)(1) = 3 > 0\)
• \(x = 1\): \((1)(-1)(3) = -3 < 0\) ✓
• \(x = 3\): \((3)(1)(5) = 15 > 0\)
Solution: \((-\infty, -2] \cup [0, 2]\)
6 Multiplicity and Sign Changes
The multiplicity of a root determines whether the polynomial changes sign at that point.
Sign CHANGES at this root
Example: \((x-2)^1\) or \((x-2)^3\)
Sign does NOT change
Example: \((x-2)^2\) or \((x-2)^4\)
Solve: \((x+1)^2(x-3) > 0\)
Roots: \(x = -1\) (mult. 2), \(x = 3\) (mult. 1)
Sign analysis:
• At \(x = -1\): even multiplicity → sign does NOT change
• At \(x = 3\): odd multiplicity → sign CHANGES
Test \(x = 0\): \((1)^2(-3) = -3 < 0\)
Test \(x = 4\): \((5)^2(1) = 25 > 0\)
Solution: \((3, \infty)\)
Once you know the sign of one interval, you can determine others by tracking sign changes at each root. Only odd multiplicity roots flip the sign!
7 Rational Inequalities
A rational inequality involves a fraction with polynomials. The critical points include both zeros of the numerator AND zeros of the denominator.
Steps for Rational Inequalities
- Move everything to one side: \(\frac{f(x)}{g(x)} \gtrless 0\)
- Find zeros of numerator (where expression = 0)
- Find zeros of denominator (where expression is undefined)
- Place ALL critical points on number line
- Test each interval and write solution (exclude denominator zeros!)
Solve: \(\frac{x-2}{x+1} \geq 0\)
Numerator = 0: \(x = 2\)
Denominator = 0: \(x = -1\) (undefined here!)
Sign chart:
Solution: \((-\infty, -1) \cup [2, \infty)\)
Note: \(x = -1\) excluded (undefined), \(x = 2\) included (≥ means = works)
Points where the denominator equals zero must ALWAYS be excluded from the solution, even if using \(\leq\) or \(\geq\).
8 Writing Solutions in Interval Notation
Express your final answer using proper interval notation with parentheses and brackets.
- Parentheses \(( )\) — endpoint NOT included (use with \(<\) or \(>\))
- Brackets \([ ]\) — endpoint IS included (use with \(\leq\) or \(\geq\))
- Union \(\cup\) — combines separate intervals
- Infinity — always use parentheses with \(\pm\infty\)
\(x < 3\) → \((-\infty, 3)\)
\(x \geq -2\) → \([-2, \infty)\)
\(-1 < x \leq 4\) → \((-1, 4]\)
\(x < -2\) or \(x> 3\) → \((-\infty, -2) \cup (3, \infty)\)
\(x \leq 1\) or \(x \geq 5\) → \((-\infty, 1] \cup [5, \infty)\)
📋 Quick Reference
\(f(x) > 0\)
Where graph is above x-axis
\(f(x) < 0\)
Where graph is below x-axis
Critical Points
Roots of numerator AND denominator
Odd Multiplicity
Crosses axis → sign changes
Even Multiplicity
Touches axis → sign stays same
Sign Chart
Test one point per interval