Algebraic Methods for Solving Nonlinear Inequalities

What Are Nonlinear Inequalities?

Nonlinear inequalities are algebraic statements where the variable appears with exponents other than 1, inside radicals, or in denominators. Unlike linear inequalities, you cannot simply isolate the variable and solve. The relationship between factors determines the solution set.

Examples include:

• Quadratic: x² - 5x + 6 > 0
• Polynomial: x³ - 4x ≤ 0
• Rational: (x + 2)/(x - 3) < 1
• With radicals: √(x + 1) > x - 1

The approach is always the same: find where the expression equals zero, then determine where it is positive or negative.

The Core Method: Sign Charts

Every nonlinear inequality reduces to a sign analysis problem. Here's the deal:

1. Move everything to one side so the inequality equals zero
2. Factor completely into linear factors
3. Identify critical points — where each factor equals zero
4. Test intervals between critical points
5. Check endpoints — include if inequality is ≥ or ≤, exclude if > or <

This works for polynomials, rational expressions, and anything you can factor.

Quadratic Inequalities

Quadratics are the most common nonlinear inequalities. You factor the quadratic, find the roots, and analyze the parabola's direction.

Standard Approach

For x² - 5x + 6 > 0:

Factor: (x - 2)(x - 3) > 0

Critical points: x = 2, x = 3

Since the parabola opens upward, it's positive outside the roots. Solution: x < 2 or x > 3

When Factoring Fails

Use the quadratic formula: x = (-b ± √(b² - 4ac)) / 2a

For 2x² + 3x - 5 ≥ 0:

Roots: x = 1 and x = -2.5

Solution: x ≤ -2.5 or x ≥ 1

Polynomial Inequalities

Higher-degree polynomials follow the same sign chart method. The key is counting sign changes in the factored form.

For x³ - x > 0:

Factor: x(x - 1)(x + 1) > 0

Critical points: -1, 0, 1

Test intervals:

• x < -1: negative × negative × negative = negative ✗
• -1 < x < 0: negative × negative × positive = positive ✓
• 0 < x < 1: positive × negative × positive = negative ✗
• x > 1: positive × positive × positive = positive ✓

Solution: -1 < x < 0 or x > 1

Rational Inequalities

Rational inequalities add one rule: never include where the denominator equals zero. That's always excluded, even with ≥ or ≤.

Method

For (x - 2)/(x + 3) ≥ 0:

1. Set numerator = 0: x = 2 (include)
2. Set denominator = 0: x = -3 (exclude)
3. Build sign chart with these critical points
4. Test intervals

Solution: x < -3 or x ≥ 2

Getting to Zero Form

When the right side isn't zero, subtract first:

(x + 1)/(x - 2) < 2

Subtract 2: (x + 1)/(x - 2) - 2 < 0

Combine: (x + 1 - 2x + 4)/(x - 2) < 0

(-x + 5)/(x - 2) < 0

Now solve the zero form.

Inequalities with Absolute Values

Absolute value inequalities require splitting into cases or using the definition:

Key Rules

• |f(x)| < a becomes -a < f(x) < a
• |f(x)| > a becomes f(x) < -a or f(x) > a

For |2x - 3| ≤ 5:

-5 ≤ 2x - 3 ≤ 5

Add 3: -2 ≤ 2x ≤ 8

Divide: -1 ≤ x ≤ 4

Splitting Method

For |x + 1| > |x - 2|:

Square both sides: (x + 1)² > (x - 2)²

x² + 2x + 1 > x² - 4x + 4

Cancel x²: 2x + 1 > -4x + 4

6x > 3

x > 1/2

Radical Inequalities

Square both sides, but only when the right side is nonnegative. Otherwise, you need to consider the domain first.

For √(x + 2) > x:

Domain: x + 2 ≥ 0, so x ≥ -2

Case 1: If x < 0, the inequality √(x + 2) > x is always true in the domain (since √(...) is always nonnegative)

Case 2: If x ≥ 0, square both sides: x + 2 > x²

x² - x - 2 < 0

(x - 2)(x + 1) < 0

Within x ≥ 0: 0 ≤ x < 2

Combine: -2 ≤ x < 2

Comparison Table: Key Methods

How to Solve Any Nonlinear Inequality

Step 1: Simplify First

Move everything to one side. Combine fractions. Get everything over a common denominator if needed.

Step 2: Factor Everything

Factor the numerator and denominator separately. If you can't factor, use the quadratic formula or synthetic division for polynomials.

Step 3: Find All Critical Points

List every value that makes any factor zero. These split the number line into intervals.

Step 4: Build the Sign Chart

Pick one test point per interval. Plug into factored form. Record positive or negative.

Step 5: Apply the Inequality

Select intervals matching your sign requirement. Add critical points back only if the inequality allows equality.

Step 6: Verify

Plug your solution endpoints back into the original inequality. Catch any domain violations.

Common Mistakes That Cost Points

• Dividing by variables — never divide by an expression containing x unless you know its sign
• Forgetting domain restrictions — denominators cannot be zero, radicands must be nonnegative
• Including excluded points — denominator zeros are never included, even with ≥ or ≤
• Losing factors when multiplying — multiplying by negative flips the inequality; multiplying by expressions with variables requires care
• Squaring without checking — squaring can introduce extraneous solutions when negative values are involved

Practice Problems

Solve: x² + x - 12 > 0

Factor: (x + 4)(x - 3) > 0

Solution: x < -4 or x > 3

Solve: (3x + 6)/(x - 1) ≤ 0

Critical points: x = -2 (numerator), x = 1 (denominator)

Solution: -2 ≤ x < 1

Solve: |3x + 1| < 7

-7 < 3x + 1 < 7

-8 < 3x < 6

Solution: -8/3 < x < 2