You finish solving x^2 - 4 > 0 and write x \in (-\infty, -2) \cup (2, \infty). You close your book. Question: is the answer right? You think so. The algebra looked clean. You flipped the sign where you were supposed to. No factor got dropped, as far as you remember.

Here is the cheapest, most reliable thirty seconds you can spend in an inequality problem: pick one number from each region of the real line and plug it into the original inequality. If the number you picked is inside your claimed solution, the inequality should hold. If it is outside your claimed solution, the inequality should fail. If either expectation is violated, something is wrong and you need to go back.

This is not double-checking the algebra — you already did that. This is testing the answer against the original question, which is a completely different check and catches completely different bugs.

The habit, in four steps

  1. Solve the inequality. Write the interval.
  2. Mark the critical points (the endpoints of your intervals) on an imaginary number line. These split \mathbb{R} into regions.
  3. Pick one convenient test value from each region — usually an easy integer like 0, \pm 1, \pm 3, or 10.
  4. Plug each test value into the original inequality (not a rearranged form). For values inside your claimed solution, the inequality should hold. For values outside, it should fail.

If any test disagrees with your answer, stop and find the bug before you move on. If all tests agree, the probability that your answer is wrong has just dropped dramatically.

Worked example: x^2 - 4 > 0

Suppose you solved this and got x \in (-\infty, -2) \cup (2, \infty). Let's sanity-check.

The critical points are -2 and 2. They split the real line into three regions: (-\infty, -2), (-2, 2), and (2, \infty). Pick one test value from each — the easiest choices are -3, 0, and 3.

Three regions of the real line for the inequality x squared minus four greater than zero, with a test point in each regionA number line running from negative six to positive six with integer ticks. The critical points at negative two and positive two are marked with hollow circles and labelled. Above the line the three regions are shaded: the left ray from negative infinity to negative two and the right ray from positive two to positive infinity are shaded in the accent colour representing the solution set, and the middle region from negative two to positive two is shaded in a neutral gray representing the excluded region. Three test points are marked with larger dots: negative three on the left ray labelled with a green check, zero in the middle region labelled with a red cross, and positive three on the right ray labelled with a green check. Each test value is annotated with the value of x squared minus four at that point: at negative three the value is five greater than zero holds, at zero the value is negative four greater than zero fails, at positive three the value is five greater than zero holds. −5 −4 −2 −1 0 1 2 3 5 x=−3: 5>0 ✓ x=0: −4>0 ✗ x=3: 5>0 ✓ inside solution excluded region inside solution Sanity-check regions for $x^2 - 4 > 0$
Three regions, three test points. The two accent-shaded rays are your claimed solution; the gray middle region is what you claimed to exclude. A green check at a test point in the accent region and a red cross at a test point in the gray region together confirm: inside holds, outside fails. That is the picture you want.

Now plug in. Test x = -3 (inside the claimed region (-\infty, -2)):

(-3)^2 - 4 = 9 - 4 = 5 > 0.\quad \checkmark

Good — the inequality holds, as expected for a point in the solution. Test x = 0 (inside the claimed excluded region (-2, 2)):

(0)^2 - 4 = -4 > 0 ?\quad \text{No — } -4 \text{ is not greater than } 0. \quad \checkmark

Also good: the inequality fails, as expected for a point you excluded. Test x = 3 (inside the claimed region (2, \infty)):

(3)^2 - 4 = 9 - 4 = 5 > 0.\quad \checkmark

All three tests agree with the claimed answer. The solution (-\infty, -2) \cup (2, \infty) passes the sanity check.

What a failed check actually looks like

Suppose a rushed student forgets the flip on a step and writes the "solution" to x^2 - 4 > 0 as x \in (-2, 2) — the opposite of the right answer. This is the kind of mistake that happens at 11pm when you are on your fourth problem and the sign-flip muscle memory is tired.

Plug x = 0, which is inside the claimed solution (-2, 2):

(0)^2 - 4 = -4 > 0?\quad \text{No.}

The test fails. A number your answer said should work, doesn't. That single failing test is a non-negotiable red flag: your answer is wrong, and you now know where to look — the step where the sign got flipped or the regions got swapped. Thirty seconds spent on a test value just saved you a wrong answer on the page.

Compare this to what happens without the sanity check: you move on, you lose the marks, and you don't even know there was a bug until the teacher returns the paper.

Why this catches bugs algebra can't

The sanity check is powerful because it is independent of the process that produced the answer. When you re-derive the solution by checking each step, you are using the same brain and often the same blind spots. If you got the sign flip wrong the first time, you will very likely get it wrong again on the re-derivation, because the mistake is built into how you see the problem.

Plugging a number is different. You are asking the original inequality a yes/no question at a specific point. The original inequality doesn't care what you think the answer is — it only cares whether the arithmetic works out. That is why the check is so reliable: the inequality itself is the arbiter.

The bugs this catches, in order of how often they actually happen on exam papers:

How to pick good test values

Cheap is good. Boring is good. You want numbers that make the arithmetic trivial so you focus on the answer, not the calculation.

When not to bother

If you are told to solve 2x > 6 and you write x > 3, you do not need to sanity-check — there is one step, and the mental cost of the plug-in is comparable to re-reading the line. Save the habit for problems where the algebra has at least two moves, where sign flips were involved, or where the answer is a union of intervals (those are the ones most prone to region-swapping).

Rule of thumb: if the answer is a union of more than one interval, or if any step involved dividing by a variable or multiplying by a negative, the sanity check is almost free insurance. Take it.

The connection to sign charts

For quadratic and polynomial inequalities, the sanity check is really a compressed version of the sign chart method: you mark the roots, and test one point in each region to decide the sign. When you use a sign chart to solve the inequality, you are already plugging a test number into each region — the sanity check is then automatic, because the chart is the check. The habit in this article is what you do when you solved the inequality some other way (say, by a pure algebraic manipulation) and want to verify independently.

For linear inequalities, a single test value from each side of the critical point is enough. For absolute-value inequalities that split into two rays, test one value from each ray and one from the gap in between. The pattern scales: one value per region, and you will sleep well at night.