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How to Solve Nonlinear Inequalities

Quadratic, polynomial, and rational inequalities all use the same technique: sign charts.

📐 Algebra II⏱️ ~15 min read📊 Advanced

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What You'll Learn

  • ✓The sign chart method (works for ALL nonlinear inequalities)
  • ✓Quadratic inequalities (x² > 4, x² - 5x + 6 ≤ 0)
  • ✓Rational inequalities with fractions
  • ✓When to include/exclude critical values

Why Can't We Just Isolate x?

With linear inequalities, you can isolate x just like equations. But with x² or x in a denominator, simply isolating doesn't work because the expression can be positive OR negative depending on x's value.

💡 The Sign Chart Method

Find where the expression equals zero or is undefined, then test each interval to see where it's positive or negative.

For more on this technique, see Math Centre's inequality guide.

The Sign Chart Method (5 Steps)

Step 1: Get Zero on One Side

Rearrange so you have expression ⪌ 0 (greater than, less than, or equal to zero).

x² > 9 → x² - 9 > 0

Step 2: Factor Completely

Factor the polynomial. For rational expressions, factor both numerator and denominator.

x² - 9 > 0 → (x+3)(x-3) > 0

Step 3: Find Critical Values

Set each factor = 0 and solve. These are your critical values.

x + 3 = 0 → x = -3

x - 3 = 0 → x = 3

Critical values: -3 and 3

Step 4: Build the Sign Chart

Plot critical values on a number line. Test a point in each interval to determine if the expression is + or −.

For (x+3)(x-3):

−−−−-3++++3++++
(test x=-4)(test x=0)(test x=4)

Step 5: Select Intervals

Choose intervals that satisfy your inequality:

  • • If > 0: pick + regions
  • • If < 0: pick − regions
  • • If ≥ or ≤: include the critical values (unless they make denominator = 0)

Worked Examples

Example 1: Quadratic Inequality

Solve: x² - 4 > 0

x² - 4 > 0
(x+2)(x-2) > 0
Critical values: x = -2, x = 2
Sign chart: (+++)[−2](−−−)[2](+++)
Want > 0, so pick + regions:
x < -2 OR x > 2
Interval notation: (−∞, −2) ∪ (2, ∞)

Example 2: Quadratic ≤ 0

Solve: x² - 5x + 6 ≤ 0

(x-2)(x-3) ≤ 0
Critical values: x = 2, x = 3
Sign chart: (+++)[2](−−−)[3](+++)
Want ≤ 0, so pick − region AND include endpoints:
2 ≤ x ≤ 3
Interval notation: [2, 3]

Example 3: Rational Inequality

Solve: (x - 1)/(x + 2) ≥ 0

(x - 1)/(x + 2) ≥ 0
Critical values: x = 1 (from numerator), x = -2 (from denominator)
Sign chart: (+++)[−2](−−−)[1](+++)
Want ≥ 0, pick + regions:
Include x = 1 (makes it = 0 ✓)
⚠️ EXCLUDE x = -2 (makes denominator = 0)
x < -2 OR x ≥ 1
Interval notation: (−∞, −2) ∪ [1, ∞)

Key Rules to Remember

✓ Include Critical Values When:

  • • Inequality has ≤ or ≥
  • • The value comes from the numerator
  • • Plugging it in gives 0 (which satisfies ≤ 0 or ≥ 0)

✗ Exclude Critical Values When:

  • • Inequality has < or > (strict)
  • • The value comes from the denominator (undefined!)

Common Mistakes to Avoid

⚠️

Including values that make denominator = 0

These values are NEVER in the solution—you can't divide by zero.

⚠️

Trying to "isolate x" like linear inequalities

For x² > 4, you can't just take square root. Use the sign chart method.

⚠️

Not getting zero on one side first

Always rearrange to expression ⪌ 0 before factoring.

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Prev: Linear InequalitiesNext: Systems of Equations
How to Solve Nonlinear Inequalities | Polynomial & Rational