Loading...

About Linear Inequations — Class 10 ICSE

Solve linear inequations in one variable and represent solution sets on the number line. This topic is part of the ICSE Class 10 mathematics syllabus (chapter: Chapter 14). On this page you can practice 60 questions across three difficulty levels — 20 easy, 20 medium, and 20 hard — each with a visual step-by-step solution, plus a timed 35-question mastery test with a live leaderboard. Worked examples and sample questions from the topic are below.

What you'll learn in Linear Inequations

  • Introduction to Linear Inequations
  • Fundamental Rules for Solving Inequations
  • Solving Complex Inequations & Replacement Sets
  • Compound Inequations and Advanced Representation
  • Summary, ICSE Exam Focus & Practice Preparation

Interactive lesson · about 15 minutes · checkpoint question after every unit

Linear Inequations — solved examples for Class 10 ICSE

Example 1easy

Consider the inequality -3x < 12. Which of the following statements correctly describes a step to solve for x?
  1. A)Divide both sides by -3, keeping the inequality sign as <.
  2. B)Divide both sides by -3, reversing the inequality sign to >.
  3. C)Add 3x to both sides, then subtract 12 from both sides, keeping the inequality sign as <.
  4. D)Multiply both sides by -1/3, keeping the inequality sign as <.

Step-by-step solution

  1. When dividing or multiplying both sides of an inequality by a negative number, the direction of the inequality sign must be reversed.
  2. In the given inequality -3x < 12, to isolate x, we need to divide by -3.
  3. Dividing both sides by -3 changes < to >:
    (−3x)/(−3)>12/(−3)(-3x)/(-3) > 12/(-3)
  4. This simplifies to x > -4.

Answer: Divide both sides by -3, reversing the inequality sign to >.

Example 2medium

Solve the inequation: 3x - 5 < 10, where x ∈ N (natural numbers).
  1. A)x < 5
  2. B)x ≤ 4
  3. C){1, 2, 3, 4}
  4. D){0, 1, 2, 3, 4}

Step-by-step solution

  1. Add 5 to both sides of the inequation:
    3x−5+5<10+5⇒3x<153x - 5 + 5 < 10 + 5 \Rightarrow 3x < 15
  2. Divide both sides by 3:
    3x/3<15/3⇒x<53x / 3 < 15 / 3 \Rightarrow x < 5
  3. Since x ∈ N (natural numbers), the possible values for x are natural numbers less than 5.
    N=1,2,3,...N = {1, 2, 3, ...}
  4. Therefore, the solution set is {1, 2, 3, 4}.

Answer: {1, 2, 3, 4}

Example 3hard

Given x ∈ Z, find the solution set of the compound inequation (2x - 5)/3 - 1 ≤ (x + 2)/2 < (x - 1)/4 + 2. Which of the following is the correct solution set?
  1. A){x | x ∈ Z, x ≤ 22}
  2. B){x | x ∈ Z, x < 3}
  3. C){x | x ∈ Z, x < 22}
  4. D){x | x ∈ Z, x ≤ 3}

Step-by-step solution

  1. First inequality: (2x - 5)/3 - 1 ≤ (x + 2)/2
  2. (2x - 5 - 3)/3 ≤ (x + 2)/2 => (2x - 8)/3 ≤ (x + 2)/2
  3. 2(2x - 8) ≤ 3(x + 2) => 4x - 16 ≤ 3x + 6 => x ≤ 22
  4. Second inequality: (x + 2)/2 < (x - 1)/4 + 2
  5. (x + 2)/2 < (x - 1 + 8)/4 => (x + 2)/2 < (x + 7)/4
  6. 2(x + 2) < x + 7 => 2x + 4 < x + 7 => x < 3
  7. Combining both conditions: x ≤ 22 and x < 3. The intersection is x < 3. Since x ∈ Z, the solution set is {x | x ∈ Z, x < 3}.

Answer: {x | x ∈ Z, x < 3}

Practice questions on Linear Inequations

  1. Q1.easy

    Solve the inequation 5x - 7 < 2x + 8, where x ∈ N (Natural Numbers). Which of the following is the correct solution set?
    1. A){1, 2, 3, 4}
    2. B){1, 2, 3, 4, 5}
    3. C){x | x < 5, x ∈ N}
    4. D){x | x ≤ 5, x ∈ N}
    Show answer

    Answer: {1, 2, 3, 4}

    Hint: First, solve the inequality algebraically as if it were an equation, then consider the replacement set.

  2. Q2.easy

    Rohan solved the inequation 2x + 3 ≥ 7, where x ∈ Z (Integers). He represented the solution set on a number line by drawing a continuous shaded line starting from a closed circle at 2 and extending to the right. What mistake did Rohan make in his representation?
    1. A)He should have used an open circle at 2 instead of a closed circle.
    2. B)He should have shaded to the left of 2 instead of to the right.
    3. C)The critical value (boundary point) of the solution should not be 2.
    4. D)For the replacement set Z, he should have marked discrete points instead of a continuous shaded line.
    Show answer

    Answer: For the replacement set Z, he should have marked discrete points instead of a continuous shaded line.

    Hint: Remember that integers are distinct, separate numbers, not a continuous range, when representing them on a number line.

  3. Q3.easy

    Solve the inequation (2x - 1)/3 + 1 > (x + 2)/2, for x ∈ R (Real Numbers). Which of the following is the correct solution?
    1. A)x < 2
    2. B)x > 2
    3. C)x < -2
    4. D)x > -2
    Show answer

    Answer: x > 2

    Hint: Find a common denominator to eliminate the fractions, then proceed with standard algebraic manipulation.

  4. Q4.medium

    Find the solution set for the inequation: 7 - 2x ≥ 3x - 13, where x ∈ Z (integers).
    1. A)x ≤ 4
    2. B)x < 4
    3. C){..., 1, 2, 3, 4}
    4. D){4, 5, 6, ...}
    Show answer

    Answer: {..., 1, 2, 3, 4}

    Hint: Gather all x terms on one side and constant terms on the other. Be careful when dealing with negative coefficients.

  5. Q5.medium

    If x ∈ R (real numbers), which of the following number lines correctly represents the solution set of 2(x - 3) + 4 ≤ 12?
    1. A)A number line with a closed circle at 7 and a shaded line extending to the left.
    2. B)A number line with an open circle at 7 and a shaded line extending to the left.
    3. C)A number line with a closed circle at 7 and a shaded line extending to the right.
    4. D)A number line with an open circle at 7 and a shaded line extending to the right.
    Show answer

    Answer: A number line with a closed circle at 7 and a shaded line extending to the left.

    Hint: Solve the inequation for x first. Remember that '≤' means the endpoint is included (closed circle) and the solution extends to smaller values.

  6. Q6.medium

    Consider the inequation -4x > 20. Which of the following statements about its solution is true?
    1. A)Dividing by -4 gives x > -5.
    2. B)Dividing by -4 gives x < -5.
    3. C)Adding 4x to both sides gives 0 > 20 + 4x.
    4. D)The solution set for x ∈ N is empty.
    Show answer

    Answer: Dividing by -4 gives x < -5.

    Hint: Recall the fundamental rule of inequations: what happens to the inequality sign when you multiply or divide by a negative number?

  7. Q7.hard

    A rectangular garden has a length that is 7 meters more than its width. If the perimeter of the garden is at least 45 meters but less than 65 meters, and the width is an integer, what is the maximum possible area of the garden?
    1. A)228 m²
    2. B)200 m²
    3. C)180 m²
    4. D)160 m²
    Show answer

    Answer: 228 m²

    Hint: Express the perimeter in terms of width (W), then set up a compound inequation for W. Find the possible integer values for W and L, and calculate the area for the largest valid width.

  8. Q8.hard

    A student was solving the inequation (7 - 2x)/5 + 3 ≥ (x - 1)/2. Their steps were as follows:
    Step 1: (7 - 2x + 15)/5 ≥ (x - 1)/2
    Step 2: (22 - 2x)/5 ≥ (x - 1)/2
    Step 3: 2(22 - 2x) ≥ 5(x - 1)
    Step 4: 44 - 4x ≥ 5x - 5
    Step 5: 44 + 5 ≥ 5x + 4x
    Step 6: 49 ≥ 9x
    Step 7: x ≥ 49/9
    Which of the following statements correctly identifies the mistake in the student's solution, if any?
    1. A)The mistake is in Step 1, where the constant 3 was added incorrectly to the numerator.
    2. B)The mistake is in Step 3, where the cross-multiplication was done incorrectly.
    3. C)The mistake is in Step 7, as the correct solution should be x ≤ 49/9.
    4. D)There is no mistake; the solution is correct.
    Show answer

    Answer: The mistake is in Step 7, as the correct solution should be x ≤ 49/9.

    Hint: Carefully review each step, especially how constant terms are combined and how the inequality sign is handled when isolating the variable.

  9. Q9.hard

    The solution set for the inequation (3x - k)/2 - 1 < x/3 for x ∈ N is {1, 2, 3, 4, 5, 6, 7}. What is the range of possible integer values for the constant k?
    1. A)12 < k ≤ 14
    2. B)13 < k ≤ 15
    3. C)14 < k ≤ 16
    4. D)15 < k ≤ 17
    Show answer

    Answer: 14 < k ≤ 16

    Hint: First, solve the inequation for x in terms of k. Then use the given natural number solution set to establish the precise bounds for the expression involving k.

These are 9 of the 60 questions available for Linear Inequations. Start practicing above to unlock visual solutions, AI coaching, and the topic leaderboard — completely free.