How To Solve Nonlinear Inequalities Step By Step?

A nonlinear inequality is an inequality where the variable is raised to a power other than one, appears in a denominator, or sits inside a radical or absolute value. To solve one, you move everything to one side, find the values that make the expression equal zero or undefined, and then test the sign of the expression in each interval those values create. The solution is the set of intervals where the inequality holds. A graphing check confirms the result.

That is the whole method. The steps below walk through it slowly, with the reasoning behind each one so the process sticks.

What Makes An Inequality Nonlinear?

An inequality is nonlinear when the expression on either side is not a straight-line relationship. If the variable only appears to the first power and is never multiplied by itself, divided into, or placed under a root, you have a linear inequality and the answer is a single interval. Nonlinear inequalities behave differently because the expression can change sign more than once.

Common nonlinear types include:

• Quadratic and higher-degree polynomial inequalities, like x² − 4 > 0 or x³ + x < 0
• Rational inequalities, where the variable is in a denominator, like (x + 1)/(x − 3) ≤ 0
• Radical inequalities, like √(x + 2) < 5
• Absolute value inequalities, like |2x − 1| ≥ 7

The key difference from linear problems is that the solution set can be split into several separate pieces. A quadratic inequality can be true on two disconnected intervals and false in the middle. Missing one of those pieces is the most common mistake.

How To Solve Nonlinear Inequalities Step By Step?

The method below works for polynomials, rational expressions, and most other nonlinear forms. It is sometimes called the critical value method or the sign chart method.

Step 1: Move everything to one side. Rewrite the inequality so that one side is zero. For example, turn x² > 4 into x² − 4 > 0. This gives you a single expression to analyze. Never divide or multiply both sides by an expression containing the variable, because you do not know whether that expression is positive or negative, and multiplying by a negative number flips the inequality sign.

Step 2: Factor the expression if you can. Factoring makes the critical values visible. x² − 4 factors into (x − 2)(x + 2). For rational expressions, factor both the numerator and the denominator.

Step 3: Find the critical values. These are the numbers that make the expression equal zero or make it undefined. Set each factor equal to zero and solve. For x² − 4, the critical values are x = 2 and x = −2. For a rational expression, also set the denominator equal to zero. Those values are never part of the solution, but they do mark boundaries where the sign can change.

Step 4: Place the critical values on a number line. This splits the number line into intervals. With critical values at −2 and 2, you get three intervals: everything less than −2, everything between −2 and 2, and everything greater than 2.

Step 5: Test one value in each interval. Pick any convenient number inside each interval and substitute it into the factored expression. You only care about the sign of the result, not the exact value. For x² − 4, test x = −3, x = 0, and x = 3. You get positive, negative, positive.

Step 6: Write the solution from the signs. Since x² − 4 > 0 requires a positive result, the solution is x < −2 or x > 2. If the inequality had been ≥ instead of >, the critical values themselves would be included, because the expression equals zero there. For a rational inequality, values that make the denominator zero are always excluded, no matter which inequality symbol is used.

Step 7: Check with a graph or a quick test. Graph the related equation and see where the curve sits above or below the x-axis. This catches sign errors fast. Picking one number from your final answer and substituting it back is a second, faster check.

How Do You Handle Rational Inequalities?

Rational inequalities follow the same six steps with one addition: the denominator’s zeros become critical values too. They are boundaries but never solutions.

Take (x + 1)/(x − 3) ≤ 0. The numerator gives a critical value at x = −1. The denominator gives one at x = 3. These split the number line into three intervals. Testing shows the expression is negative between −1 and 3, positive outside that range.

Because the inequality asks for values less than or equal to zero, you might be tempted to include x = 3. You cannot. At x = 3 the expression is undefined, so it has no value there at all. The solution is −1 ≤ x < 3, with a closed bracket at −1 and an open one at 3. That distinction between a zero of the numerator and a zero of the denominator is the single most important detail in rational inequalities.

How Do Absolute Value And Radical Inequalities Work?

Absolute value inequalities split into two cases based on a simple rule. If |A| < b, then A is between −b and b. If |A| > b, then A is either less than −b or greater than b. For |2x − 1| ≥ 7, you solve 2x − 1 ≤ −7 and 2x − 1 ≥ 7 separately, then combine. The answer is x ≤ −3 or x ≥ 4.

Radical inequalities need one extra check: the expression under the root must be non-negative, or the root is not a real number. For √(x + 2) < 5, you first require x + 2 ≥ 0, so x ≥ −2. Then square both sides to get x + 2 < 25, or x < 23. Combining both conditions gives −2 ≤ x < 23. Squaring can introduce values that do not actually satisfy the original inequality, so always test your answer in the original form.

What Are The Most Common Mistakes?

Most errors in these problems come from a small set of habits. Knowing them in advance prevents most of them.

• Multiplying both sides by an expression with a variable in it. This is invalid unless you know the sign of that expression, and you usually do not.
• Forgetting that denominator zeros are excluded. They are boundaries, never solutions.
• Testing only one interval. Every interval between critical values needs its own test.
• Assuming the sign alternates every time. It usually does for simple polynomials, but not always, and not for every type of expression.
• Dropping the domain restriction on a radical. The expression under the root must be non-negative.
• Squaring both sides without checking. Squaring is not a reversible operation, so it can create false solutions.

One detail worth knowing: for a polynomial with distinct real roots, the sign does alternate from interval to interval. That pattern is real, not a coincidence. But it breaks down when a factor is repeated an even number of times, like (x − 1)², which never changes sign. Relying on the alternation shortcut without checking for repeated factors is a common source of wrong answers.

When Should You Use A Sign Chart Or A Graph?

A sign chart is the most reliable tool for these problems, especially for rational expressions where a graph can be hard to read near a vertical asymptote. Build the chart by listing the critical values in order and marking the sign of each factor across each interval. Multiply the signs to get the overall sign.

A graph is faster for a quick check and for building intuition about where the expression is positive or negative. For a polynomial, the graph crosses the x-axis at each critical value, and the solution is wherever the curve is above or below the axis, depending on the inequality. For a rational function, the graph has a vertical asymptote at each denominator zero, and the curve approaches but never touches it.

Use both when you can. The chart gives you the exact answer. The graph tells you whether that answer makes sense.