How to Solve CAT 2026 Inequality, Modulus & GIF Questions?

You start a mock test and the screen shows an inequality question. Your head immediately goes to the worry: “Will the sign change here?” A few minutes pass. Then ten minutes. By the time you fill half a page with rough steps, you still feel unsure.

If this has happened to you before, you are not alone. A lot of the CAT 2026 Quant Questions people discuss most often link back to three areas that look simple at first. Inequalities, modulus, and the greatest integer function, or GIF.

You do not need many long formulas. What matters is the way you work. You need a few good habits and some quick checks.

In this guide, we will cover each topic in order. You will see worked examples, the usual mistakes to watch for, and a practice set at the end. Keep a notebook close. Try each example once before you look at the solution.

Why focus on these three topics

Algebra is one of the steady parts of CAT Quantitative Aptitude Questions. Inequalities, modulus, and GIF are right in the middle of it. There are clear reasons to spend your time on them in exam conditions.

They're short but tricky. But they are not. Most of the time, the problem is only two lines. Still, the key part sits in those two lines.
They reward understanding, not memory. Not if you only memorize. With a modulus problem, you cannot just cram. You need to understand the idea behind it.
These show up a lot as TITA problems. There are no answer choices to lean on. Also, there is no cut for wrong guesses. If you are sure, you can take marks that others miss.
They combine with each other. A modulus inside an inequality is a common trick. So is a GIF put inside a modulus. This is the sort of CAT move you should watch for.

The good part is this. The three areas use methods that repeat. You can learn the pattern and use it again. Now, let’s begin with the inequality set.

CAT 2026 Inequality Questions: Ways to Keep Marks

Most errors in CAT 2026 Inequality Questions come from a few bad habits. Address these first.

If you multiply or divide by a negative, the inequality sign changes. From −2x > 6, you end up with x < −3.
Do not cross-multiply if you are not sure what sign the denominator has. That mistake shows up a lot in rational inequalities.
When you square, only do it if both sides are already at least zero. If not, you might remove valid values or add values that should not work.
Also watch out for fractions and reciprocals. When both numbers are positive, a > b leads to 1/a < 1/b. If the signs are mixed, that rule no longer holds.

The Wavy Curve Method

If you run into a product, or a fraction made from linear factors, use the wavy curve method right away.

Solve Example: (x − 1)(x + 3) / (x − 2) ≤ 0.

First, list the points that matter.The numerator is zero at x = −3 and x = 1.The denominator is zero at x = 2, so x = 2 is not allowed.
Check intervals from right to left. For x > 2, (x − 1) is positive and (x + 3) is positive, and the denominator is positive too. So the fraction is above 0 there.
Now move left past each critical point. Between 1 and 2, the sign flips, and the expression is below 0. Between −3 and 1, it flips again, and the expression is above 0. For x < −3, it flips once more, and the expression is below 0.
We need the expression to be 0 or negative. Include x = −3 and x = 1 since those make the numerator 0. Exclude x = 2 since the denominator becomes 0.

Final answer: x ≤ −3 or 1 ≤ x < 2.

Quick reminder: if a factor shows up twice, like (x − 4)², it does not change the sign at that point. Also, when the inequality says “equal to,” those endpoint values can still work.

The Cross Multiplication Pitfall

Example problem: Solve x / (x − 2) > 3.

A lot of students do this: they multiply both sides by (x − 2) and get x > 3x − 6. Then they rearrange and end up with x < 3. But that step is not valid. The problem is that (x − 2) might be less than 0, and then the inequality would flip.

Here is a quick check. Pick x = 0. It fits x < 3. Yet 0 / (−2) equals 0, and 0 is not bigger than 3.

A safer approach is to move terms to one side and keep the fraction intact:

x / (x − 2) − 3 > 0

(x − 3x + 6) / (x − 2) > 0

(6 − 2x) / (x − 2) > 0

(x − 3) / (x − 2) < 0

Next, use the sign chart with the wavy line setup. The solution comes out to 2 < x < 3.

Test x = 2.5: you get 5, and 5 is greater than 3.

AM GM for Max and Min Questions

If a and b are positive, then (a + b) / 2 ≥ √(ab). The equality case happens only when a equals b.

Example: Find the minimum of x + 16/x for x > 0.

Since x is positive, you can use AM-GM.

AM-GM says x + 16/x ≥ 2√(x · 16/x) = 2√16 = 8. Equality happens when the two parts match, so x = 4. So the smallest value is 8.

Quick checks you should do first.

Make sure all terms are positive when the rule needs it. Also make sure the equality case is actually possible in the given domain. If equality cannot happen, then the result is only a lower or upper bound. It may not be the true minimum.

Your fast safety net (about 20 seconds)

After you finish an inequality, take one test value from each region you got. Put it back into the original inequality. This often catches a wrong sign right away.

CAT 2026 modulus questions: focus on distance

Think of |x − a| as the gap between x and a on the line.

For instance, |x − 3| < 2 means x stays within 2 of 3.

That gives 1 < x < 5.

Also, |x − 3| > 2 means x is more than 2 away from 3.

So x < 1 or x > 5.

Once you get this picture, you may solve it mentally.

Your friend may still be trying to read the question fully.

The Case Method: a quick way to handle it

If an absolute value setup has more than one modulus, cut the number line at the break points.

Example: Solve |x − 2| + |x + 3| = 7.

First, find the break points. They come from the inside parts: x − 2 = 0 gives x = 2 x + 3 = 0 gives x = −3 So the line splits into three parts.

x < −3 Then |x − 2| = 2 − x and |x + 3| = −x − 3 So the left side is (2 − x) + (−x − 3) = −2x − 1 Set −2x − 1 = 7 That gives x = −4 Check: −4 is less than −3, so it works.
−3 ≤ x ≤ 2 Now |x − 2| = 2 − x and |x + 3| = x + 3 Left side: (2 − x) + (x + 3) = 5 This would mean 5 = 7 No solution in this range.
x > 2 Now |x − 2| = x − 2 and |x + 3| = x + 3 Left side: (x − 2) + (x + 3) = 2x + 1 Set 2x + 1 = 7 That gives x = 3 And 3 is bigger than 2, so it works.

Final answer: x = −4 or x = 3.

When you solve an absolute value equation, do not stop after one pass.

Always test for extra cases.

Example: Solve |x − 3| = 2x − 1.

Step 1: Note that |something| is never below zero. So the expression on the right must be at least zero.

Make sure 2x − 1 ≥ 0.

Step 2: Split into two cases.

Case 1: x − 3 = 2x − 1. This gives x = −2. Now check the right side: 2(−2) − 1 = −5.That is negative. So discard this case.
Case 2: 3 − x = 2x − 1. This gives x = 4/3. Check again: 2(4/3) − 1 = 5/3, which is not negative. Also verify the absolute value: |4/3 − 3| = 5/3. Works.

Final answer: x = 4/3.

If you do not run the sign check, you may keep both cases and get a wrong extra solution. Then CAT can give zero for that careless slip.

Minimum using distances

Check the expression |x − 2| + |x + 3|.

It adds the distance from x to 2 and the distance from x to −3.

The smallest it can be is the gap between the two fixed points.

That gap is 5.

You get this same minimum for any x from −3 up to 2.

Now for three points. The lowest sum lands at the middle point.

That middle point is the median.

Example. Minimize |x − 1| + |x − 4| + |x − 9|.

The median of 1, 4, 9 is 4.

Plug in x = 4: |4 − 1| + |4 − 4| + |4 − 9| = 3 + 0 + 5 = 8.

You do not need calculus.

You do not need to split into cases either.

Another fact to keep: |a + b| ≤ |a| + |b|.

You get equality when a and b share the same sign, or if one of them is zero.

CAT often uses this for questions that ask for the count of integer values.

CAT 2026 GIF Questions: Split the Number Into Two Parts

Let [x] be the greatest integer that is less than or equal to x.

The fractional part is {x} = x − [x].

It always falls between 0 and 1, but it never reaches 1.

Negative values are where many mistakes show up, so be careful:

[3.7] = 3 and {3.7} = 0.7
[−3.7] = −4 and {−3.7} = 0.3

You will use these rules more than once:

x = [x] + {x}
If n is an integer, then [x + n] = [x] + n
If x is an integer, then [−x] = 0. Otherwise, [−x] = −1

Treat [x] as a Single Unknown

Example: Solve [x]² − 5[x] + 6 = 0.

Set [x] = k. Then k² − 5k + 6 = 0. From this, k = 2 or k = 3. So [x] can be 2 or 3. That means 2 ≤ x < 4.

Pay attention: the result is an interval. It is not only x = 2 and x = 3. Many people miss that part.

The x = n + f way

When you see [x] and {x} in the same problem, rewrite x as x = n + f.

Here n is an integer, and 0 ≤ f < 1.

The limits on f are what restrict n.

Example: Solve [x] + 2{x} = 5.4.

Let [x] = n and {x} = f.

Then n + 2f = 5.4.

Since 0 ≤ f < 1, you have 0 ≤ 2f < 2. So n must fit between 3.4 and 5.4.

That leaves n = 4 or n = 5.

If n = 4, then 2f = 1.4, so f = 0.7. Then x = 4 + 0.7 = 4.7.
If n = 5, then 2f = 0.4, so f = 0.2. Then x = 5 + 0.2 = 5.2.

You can check both values directly.

GIF in number theory

To find the top power of a prime p inside n!, you add up these terms:

[n/p] + [n/p²] + [n/p³] + ...

Example: power of 3 in 50!

[50/3] + [50/9] + [50/27]

= 16 + 5 + 1

= 22

GIF and an inequality

Take the case [x]² < 9.

Since 9 is 3², you get −3 < [x] < 3.

So [x] can be −2, −1, 0, 1, or 2.

That gives −2 ≤ x < 3.

You can see the pattern: the left side allows −2, but 3 is not allowed on the right.

Mistakes That Keep Costing Marks

Before you start the practice, check this list.

These are the errors students make again and again:

When you divide, you may flip the sign by mistake. Positive means no flip. Negative means you must flip.
Do not cancel a factor such as (x − 2) on both sides without thinking. Its sign can change what works.
If a fraction has a denominator, make sure the denominator is not zero. Exclude those x values.
In modulus questions, still test which x values are valid.
Remember that [−3.7] is −4, not −3.
If the correct range is 2 ≤ x < 4, do not stop at only x = 2 or x = 3.
Avoid using AM-GM when the numbers are negative.

A Smart CAT 2026 Quant Strategy for These Topics

Good CAT 2026 Quant prep is not about grinding more and more questions. It is about using each question to learn.

Build a one-page rule sheet. Put the key sign rules there. Add the steps for the wavy curve idea. Note what modulus changes mean in distance terms. Also write the GIF related properties in your own simple words. Read this page before you start practice every day.
Practice by topic first. Take one topic at a time. Do about 15 to 20 questions in one sitting. After a few rounds, the pattern starts to feel normal.
Then mix them. Once you feel ready with the method, do mixed sets with a timer. In CAT, you do not get labels for what topic each question comes from.
Maintain an error log. For every wrong answer, write the reason. Use short tags like concept gaps, careless mistakes, or time pressure. Check the note after a few days. You will see repeated problems.
Use plug-in values in MCQs. When options are ranges, test a single value inside each range. This can save time when you just need to eliminate options.
Set a time cap. If you are stuck for around two minutes, move on. Mark it and return later if you have time. One long struggle should not block many easy questions.
Redo mock mistakes a few days later. A few days after a mock, solve the same missed questions again. Try a different approach if you can. That is how the fix becomes real.

If you are chasing a high rank in CAT 2026 Quant, add one last habit. After solving, ask, “Can I do this faster?” The median approach in modulus and the wavy curve method for inequalities can give quick speedups.

CAT 2026 Quant Practice Set

Try these first. Do not peek yet.

Count integers x in the range −6 ≤ x ≤ 6 that make (x+3)(x-4)(x-1) ≤ 0?
Get the lowest value of |x-5|+|x+1|+|x-9|
Add all real solutions of |x-1|=2|x+2|
Given [x]=3 and [y]=-2, find the possible values of x-y.
Find the biggest power of 7 that divides 200!.
How many real x satisfy {x}=x/5?

Answers and Quick Hints

7. The solution set is x ≤ −3 or 1 < x ≤ 4. Within the given range, that's −6, −5, −4, −3, 2, 3 and 4.
10. The median of −1, 5 and 9 is 5. At x = 5: 0 + 6 + 4 = 10.
−6. Solving x − 1 = 2(x + 2) gives x = −5, and x − 1 = −2(x + 2) gives x = −1. Both check out, and their sum is −6.
4 < x − y < 6. Here x lies in [3, 4) and y lies in [−2, −1), so x − y is strictly between 4 and 6.
32. [200/7] + [200/49] = 28 + 4.
4. Write x − [x] = x/5, so [x] = 4x/5. This means x = 5n/4 for an integer n, and n ≤ 5n/4 < n + 1 holds only for n = 0, 1, 2, 3. The values are x = 0, 1.25, 2.5 and 3.75.

Score yourself honestly. If you got four or more correct without looking, you are in a good place. If you did not, that is fine too. Return to the part where you made the mistake. Redo the worked example. Then attempt the question again after a day.

Final thoughts: small topics, steady confidence.

Inequalities, modulus, and GIF do not sound exciting. Still, these are areas where preparation pays off fast. A focused student picks up marks. A shaky student drops them just as quickly. It is not usually pure talent. It comes from how you work. It also depends on practice and quick feedback on what you miss.

That is why good guidance matters. At TCM Education, we push for clear ideas, practice by topic, and mock tests that you review properly. We also want mentors to explain the reasons behind each shortcut. You can use the same approach in the quant parts of IPMAT, CUET, and other entrance exams.

If you want a clear plan for your CAT 2026 prep, check what TCM Education has and decide if it matches your style. For now, keep practising. Write down your mistakes in an error log. Each trap you learn to catch becomes a mark you do not lose.

Frequently Asked Questions

1. Why do CAT 2026 Quant questions on inequalities, modulus, and GIF seem hard?
They look simple at first. But each one has a catch. Sometimes the sign changes in the middle. Sometimes a step gives a value that does not pass the final test. In GIF, a negative value can cause trouble. When you learn what to watch for in each type, the questions feel faster to solve.

2. What is the quickest way to handle CAT 2026 inequality questions?
When the form has products or fractions of linear factors, use the curve method. First, note the key points. Then test the sign in the far right interval. Move to the left and see the sign alternate. Finally, choose the intervals that fit the given inequality.

3. Why should I not cross-multiply in rational inequalities?
You might multiply by something that is positive, or you might multiply by something negative. If it is negative, the inequality flips. To avoid this, move all parts to one side. Put them into one fraction. Then work with the curve method.

4. What is the easiest way to view CAT 2026 modulus questions?
Read |x − a| as the distance from x to a on the number line. So |x − 3| < 2 means x is less than 2 units away from 3. This picture often removes the need for many case splits.

5. How do I stop making mistakes in modulus equations?
Check every answer by putting it back into the original equation. A modulus value cannot be negative. So if a candidate makes the other side negative, throw that candidate out.

6. What is the smallest value of a sum of absolute values, and is there an easy way?
For |x − a| + |x − b|, the minimum is the gap between a and b. For three or more points, the minimum happens at the median of the given numbers. So for |x − 1| + |x − 4| + |x − 9|, the best x is 4, and the minimum total is 8.

7. How does the greatest integer function behave when x is negative?
[x] means the greatest integer that is not more than x. So [−3.7] equals −4, not −3. The fractional part is {x} = x − [x]. It always lands in the range from 0 up to 1. Hence {−3.7} is 0.3.

8. How do I handle problems that include both [x] and {x}?
Start by writing x = n + f. Here n is an integer and f is a number with 0 ≤ f < 1. Once you use this form, the condition on f restricts which n can work. Then you test each allowed n and solve for f.

9. Are these usually multiple choice questions or TITA?
They can be either. If it is a TITA question, there is no penalty for a wrong guess. So if you can reach a valid approach, it is worth attempting.

10. How can TCM Education support my CAT 2026 Quant prep?
TCM Education teaches the ideas in a simple way. You also get practice by topic, plus mock review. The mentors guide you through why a shortcut works, not just how to apply it. This style helps you form steady methods for the exam. It also helps in the quant parts of IPMAT, CUET, and other entrance tests.