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Quantitative Aptitude · Linear Inequalities

Inequalities and Their Types for CA Foundation

Updated 1 October 2026 · Fact-checked

An inequality compares two expressions using <, >, ≤ or ≥ instead of =. It can be strict (< or >) or slack (≤ or ≥), and linear (highest power 1) or quadratic (highest power 2). To solve MCQs, identify the symbol, the degree, then test the given values.

Understand Inequalities and Their Types

An equation says two things are equal. An inequality says one thing is smaller or larger than another. For example, 3x + 2 > 11 is true for many values of x, not just one. So the answer to an inequality is usually a range of values, not a single number.

Four symbols are used:
- < means less than
- > means greater than
- ≤ means less than or equal to
- ≥ means greater than or equal to

A strict inequality uses < or >. The boundary value is not included. In x > 3, the value 3 does not satisfy it. A slack inequality (also called non-strict) uses ≤ or ≥. The boundary value is included. In x ≥ 3, the value 3 does satisfy it.

Inequalities are also classified by degree. A linear inequality has the variable only to power 1, such as 2x - 5 < 7 or 3x + 2y ≤ 12. A quadratic inequality has the highest power 2, such as x² - 5x + 6 > 0. At Foundation level, the chapter mainly solves linear inequalities.

By number of variables, an inequality is in one variable (like x + 4 ≤ 9) or two variables (like x + y ≥ 5). One-variable solutions are shown on a number line. Two-variable solutions are shown as a region on a graph.

A double inequality such as 2 < x ≤ 7 says x lies between two values. Here 2 is excluded and 7 is included. Questions often test whether you can read the endpoints correctly.

Key formulas to remember

Strict inequalities
a < b or a > b
Equality is not allowed. The boundary value is excluded, shown by an open circle on the number line.
Slack inequalities
a ≤ b or a ≥ b
Equality is allowed. The boundary value is included, shown by a filled circle.
Linear inequality in one variable
ax + b < 0 (or >, ≤, ≥), where a ≠ 0
Highest power of x is 1.
Quadratic inequality in one variable
ax² + bx + c < 0 (or >, ≤, ≥), where a ≠ 0
Highest power of x is 2.
Linear inequality in two variables
ax + by ≤ c (or <, >, ≥)
Its solution is a region of the plane, not a single point.
Double inequality
p < x ≤ q
Means x > p and x ≤ q together. Check each end's symbol separately.

How to solve Inequalities and Their Types questions

Use this method for any question that asks you to classify an inequality or check whether a value satisfies it.

  1. 1Read the symbol carefully. Note whether it is <, >, ≤ or ≥.
  2. 2Decide strict or slack. < and > are strict. ≤ and ≥ are slack.
  3. 3Find the highest power of the variable. Power 1 means linear. Power 2 means quadratic.
  4. 4Count the variables. One variable gives a range on a number line. Two variables give a region.
  5. 5If a value is given, substitute it into the inequality and simplify.
  6. 6Check the boundary value separately. It passes only for a slack inequality.
  7. 7Match your result with the options and pick the one that fits all conditions.

Quickest way: Substitute and eliminate

When to use it: Use this when the MCQ gives numbers or ranges as options and you must find which satisfies the inequality.

  1. Test the boundary value first. It removes options quickly, since strict inequalities reject it.
  2. Test one value clearly inside the range and one clearly outside.
  3. Cross out options that fail either test.
  4. For classification, look only at the symbol and the highest power. Ignore the rest of the expression.
  5. Skip a question if the algebra gets long. A wrong answer costs 0.25 marks.

Common mistakes in Inequalities and Their Types

  • Treating x > 3 as if 3 is a solution.

    Students mix up strict and slack symbols.

    Fix: Remember: only ≤ and ≥ include the boundary. Substitute 3 and see if 3 > 3 is true. It is not.

  • Calling x² + 3x ≥ 0 a linear inequality because it looks simple.

    Students see only a few simple terms, or notice the 3x term, and overlook the x² term that sets the degree.

    Fix: Always find the highest power of the variable. Power 2 means quadratic.

  • Calling 3x + 2y ≤ 6 a quadratic inequality because it has two variables.

    Students confuse number of variables with degree.

    Fix: Degree is the highest power. Here every variable has power 1, so it is linear in two variables.

  • Reading 2 < x ≤ 7 as including 2 and excluding 7.

    Students ignore which symbol sits at which end.

    Fix: Read each end separately. < at 2 excludes 2. ≤ at 7 includes 7.

  • Thinking an inequality has one answer like an equation.

    Habit from solving equations.

    Fix: An inequality usually has infinitely many solutions forming a range. Check whether an option is in the range.

Worked examples

Example 1

Which of the following is a strict linear inequality in one variable? (a) 2x + 3 ≤ 9 (b) x² - 4 > 0 (c) 5x - 1 > 14 (d) x + y < 6

Show the solution
  1. (a) uses ≤, so it is slack. Eliminate.
  2. (b) has x², so it is quadratic. Eliminate.
  3. (c) uses > so it is strict. The highest power of x is 1 and there is one variable. It fits.
  4. (d) is linear and strict but has two variables. Eliminate.

Answer: (c) 5x - 1 > 14

Example 2

Which value of x satisfies the slack inequality 3x - 2 ≥ 10? (a) 3 (b) 4 (c) 2 (d) 0

Show the solution
  1. Solve: 3x ≥ 12, so x ≥ 4.
  2. Test the boundary 4: 3(4) - 2 = 10, and 10 ≥ 10 is true.
  3. Test 3: 3(3) - 2 = 7, which is less than 10. Fails.
  4. Test 2 and 0: they give 4 and -2, both less than 10. Fail.

Answer: (b) 4

Example 3

For the double inequality 2 < x ≤ 7, which integer is NOT a solution? (a) 3 (b) 7 (c) 2 (d) 5

Show the solution
  1. The left end uses <, so x = 2 is excluded.
  2. The right end uses ≤, so x = 7 is included.
  3. 3 and 5 lie strictly between 2 and 7, so they satisfy it.
  4. 7 satisfies it. 2 does not, since 2 < 2 is false.

Answer: (c) 2

Exam tips

  • Questions on this topic are usually quick classification or substitution checks. Do them fast and save time for longer sums.
  • Always test the boundary value when the options include it. This is the most common trap.
  • Identify degree by the highest power, not by how complex the expression looks.
  • If two options differ only in an open or closed endpoint, the strict or slack symbol decides the answer.

Practice questions from Linear Inequalities

Inequalities and Their Types: frequently asked questions

What is a linear inequality in one variable?

It is an inequality where the variable has power 1 and there is only one variable, such as 4x - 3 < 9. Its solution is a range of values on a number line.

What is the difference between strict and slack inequality?

A strict inequality uses < or > and excludes the boundary value. A slack inequality uses ≤ or ≥ and includes the boundary value. For example, x ≥ 5 accepts 5 but x > 5 does not.

How is a quadratic inequality different from a linear one?

A quadratic inequality has the variable raised to the power 2, like x² - 9 < 0. A linear inequality has the highest power 1. Quadratic solutions can split into two separate ranges, while linear ones give a single range.

Is this topic important for CA Foundation Paper 3?

Linear Inequalities is part of Business Mathematics in Paper 3, which is an MCQ paper with 0.25 negative marking. The basics here support every later topic in the chapter.