Linear and Quadratic Inequalities ISC Class 11: Method of Intervals

What Linear and Quadratic Inequalities is about

Weightage: Linear and Quadratic Inequalities belongs to the unit Algebra, which CISCE weights at 26 of the 80 marks in the ISC Class XI Mathematics Paper I (Theory). Class XI is examined by the school, so your school sets the paper within this weightage.Source: CISCE ISC Mathematics syllabus, examination year 2028, Class XI unit weightage

Key concepts

Rules for working with inequalities

If a < b, then: a + c < b + c and a − c < b − c for any real c; ac < bc if c > 0; but ac > bc if c < 0 (the sign reverses). The same holds for division. Also, if 0 < a < b then 1/a > 1/b (taking reciprocals of positives reverses the sign). Squaring is safe only when both sides are known to be non-negative. These rules let you move terms and clear fractions, but every time you multiply or divide by a negative quantity, you must flip the sign.

Worked example

Solve −3x + 7 > 1.

Try it first, then show the solutionSolution

1. Subtract 7: −3x > −6
2. Divide by −3 and reverse the sign: x < 2

Answer: x < 2, i.e. (−∞, 2)

Linear inequalities and the number line

A linear inequality in one variable has the form ax + b < 0 (or >, ≤, ≥) with a ≠ 0. Clear fractions by multiplying by the (positive) LCM, collect x-terms on one side and divide. Show the answer on a number line: a filled dot for ≤ or ≥, an open dot for < or >, and an arrow or bar for the solution set. Interval notation: (a, b) excludes both ends, [a, b] includes both, and ±∞ always takes a round bracket. If the variable is restricted (for example x a natural number), list only the allowed values.

Worked example

Solve 3(x − 2) + 5 ≤ 2x + 7 and show the solution on a number line.

Try it first, then show the solutionSolution

1. 3x − 6 + 5 ≤ 2x + 7, so 3x − 1 ≤ 2x + 7
2. x ≤ 8
3. Number line: filled dot at 8, shading to the left

Answer: x ≤ 8, i.e. (−∞, 8]

Double and simultaneous linear inequalities

A double inequality such as p < ax + b ≤ q is solved by doing the same operation to all three parts at once. For a system of inequalities, solve each one separately and then take the intersection (the values satisfying all of them); drawing each solution on the same number line makes the overlap easy to see. If the solution sets do not overlap, the system has no solution.

Worked example

Solve −5 < (2x − 3)/3 ≤ 4.

Try it first, then show the solutionSolution

1. Multiply all parts by 3: −15 < 2x − 3 ≤ 12
2. Add 3: −12 < 2x ≤ 15
3. Divide by 2: −6 < x ≤ 7.5

Answer: (−6, 7.5]

Quadratic inequalities by the method of intervals

To solve ax² + bx + c > 0 (or <, ≥, ≤): (1) make a > 0, multiplying by −1 and flipping the sign if needed; (2) factorise as a(x − α)(x − β) with α < β; (3) mark α and β on a number line, giving three intervals; (4) the expression is positive outside the roots and negative between them. So (x − α)(x − β) < 0 ⇔ α < x < β and (x − α)(x − β) > 0 ⇔ x < α or x > β. Include the end-points for ≤ or ≥. If you are unsure, test one value in each interval.

Worked example

Solve 2x² − 5x − 3 < 0.

Try it first, then show the solutionSolution

1. 2x² − 5x − 3 = (2x + 1)(x − 3); critical points −1/2 and 3
2. a > 0, so the expression is negative between the roots

Answer: −1/2 < x < 3

Perfect squares and quadratics with no real roots

If the quadratic is a perfect square, (x − k)² ≥ 0 for every x and equals 0 only at x = k. So (x − k)² ≥ 0: all real x; (x − k)² > 0: all real x except k; (x − k)² ≤ 0: only x = k; (x − k)² < 0: no solution. If a > 0 and D < 0, the quadratic is positive for all real x (it never crosses the axis), so ax² + bx + c > 0 holds for every x and ax² + bx + c < 0 has no solution.

Worked example

Solve (i) x² − 6x + 9 > 0 (ii) x² − 8x + 16 ≤ 0 (iii) x² + 2x + 5 < 0.

Try it first, then show the solutionSolution

1. (i) (x − 3)² > 0: true unless x = 3
2. (ii) (x − 4)² ≤ 0: only when x = 4
3. (iii) D = 4 − 20 < 0 and a > 0, so x² + 2x + 5 is always positive

Answer: (i) x ∈ ℝ, x ≠ 3 (ii) x = 4 (iii) no real solution

Rational inequalities f(x)/g(x) ≤ a

Never cross-multiply, because g(x) may be negative for some x and positive for others. Instead: (1) subtract a, so the right side is 0; (2) combine into a single fraction N(x)/D(x); (3) find the zeros of N and D (the critical points); (4) test the sign in each interval; (5) include zeros of N for ≤ or ≥, but always exclude zeros of D, where the expression is undefined. Note that N/D and N·D have the same sign wherever D ≠ 0.

Worked example

Solve (x + 1)/(x − 2) ≥ 3.

Try it first, then show the solutionSolution

1. (x + 1)/(x − 2) − 3 ≥ 0 ⇒ (x + 1 − 3x + 6)/(x − 2) ≥ 0 ⇒ (7 − 2x)/(x − 2) ≥ 0
2. Multiply by −1 and flip: (2x − 7)/(x − 2) ≤ 0; critical points 2 and 7/2
3. The fraction is negative between them; include 7/2, exclude 2

Answer: 2 < x ≤ 7/2

Repeated factors in the method of intervals

When the expression is a product of several linear factors, the sign changes at each critical point that comes from an odd power and does not change at one from an even power. Start from the right-most interval, where every factor (x − k) is positive, and move left, switching sign only at odd-power points. An even-power point is still a zero, so it is included for ≥ or ≤ and excluded for > or <.

Worked example

Solve (x − 1)²(x + 2)(x − 3) < 0.

Try it first, then show the solutionSolution

1. Critical points −2, 1, 3; (x − 1)² does not change sign
2. Signs from the right: x > 3: +; 1 < x < 3: −; −2 < x < 1: − (no change at 1); x < −2: +
3. Strict inequality, so x = 1 (a zero) is excluded

Answer: −2 < x < 3, x ≠ 1

Maths in real life: Linear and Quadratic Inequalities

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Topics in this chapter

• Linear Inequalities in One Variable and Modulus
• Quadratic and Rational Inequalities: Method of Intervals

Linear and Quadratic Inequalities formula sheet

Adding a constant

$a<b\Rightarrow a+c<b+c$

True for every real c.

Multiplying by a negative

$a<b,c<0\Rightarrow ac>bc$

Sign reverses.

Between the roots

$(x-\alpha )(x-\beta )<0⟺\alpha <x<\beta (\alpha <\beta )$

Use ≤ and closed ends for ≤ 0.

Outside the roots

$(x-\alpha )(x-\beta )>0⟺x<\alpha \text{ or }x>\beta$

Two separate intervals.

Always positive

$a{x}^2+bx+c>0\forall x\in \mathbb{R}⟺a>0,D<0$

Always negative: a < 0, D < 0.

Rational inequality

$\frac{N(x)}{D(x)}\ge 0⟺N(x)D(x)\ge 0,D(x)\ne 0$

Bring everything to one side first.

Linear and Quadratic Inequalities: diagrams to practise

Practise each one on paper: draw it from the checklist, then tick off every label.

How the school exam asks Linear and Quadratic Inequalities

The question patterns to expect, and the method that works for each.

Tricks to remember Linear and Quadratic Inequalities

Short memory aids. Say each one aloud once, then use it on a question the same day.

• Negative flips

  Multiply or divide by a negative: flip the sign, every time.

  Use it when: Linear inequalities.

• Outside same, inside opposite

  A quadratic has the sign of a outside its roots and the opposite sign between them.

  Use it when: Quadratic inequalities.

• Never cross-multiply

  With an unknown denominator, bring everything to one side instead.

  Use it when: Rational inequalities.

• Denominator zeros are always out

  Open circle on every zero of the denominator, even for ≤ or ≥.

  Use it when: Writing the final answer.

• Even powers bounce

  The sign curve bounces at a squared factor and crosses at a single one.

  Use it when: Repeated factors.

Linear and Quadratic Inequalities: exam-level solved questions

Ordered from exam level to HOTS. Cover the solution, write your own, then compare step by step.

Case-study practice (Section D style)

Read the situation, then answer every part. In a school exam set on the board pattern this is a 5-mark question split into 1 + 1 + 1 + 2 marks.

Where the marks are

How a typical answer is split into step marks. Show every step: a right final answer alone rarely earns full marks.

Where students lose marks in Linear and Quadratic Inequalities

The slips examiners see most often, and what to write instead.

• ✗Mistake: Not reversing the sign

  Students divide by a negative coefficient as if it were positive.

  Wrong working

  −4x ≥ 12 ⇒ x ≥ −3.

  Right working

  −4x ≥ 12 ⇒ x ≤ −3.

  ✓Do this: Circle every negative multiplier before you divide.

  Usually costs: Full marks for the part.

• ✗Mistake: Cross-multiplying a rational inequality

  The denominator can be negative, which would reverse the sign for some x.

  Wrong working

  (x + 3)/(x − 1) ≤ 2 ⇒ x + 3 ≤ 2x − 2 ⇒ x ≥ 5.

  Right working

  One side zero and a sign chart give x < 1 or x ≥ 5.

  ✓Do this: Subtract, combine, sign chart.

  Usually costs: 2-3 marks.

• ✗Mistake: Including a zero of the denominator

  Students copy ≥ to both critical points.

  Wrong working

  (x − 5)/(x − 1) ≥ 0 ⇒ x ≤ 1 or x ≥ 5.

  Right working

  x = 1 makes the fraction undefined: x < 1 or x ≥ 5.

  ✓Do this: Mark denominator zeros with open dots first.

  Usually costs: 1 mark.

• ✗Mistake: Writing an impossible chain

  'x < −3 or x > 2' is written as −3 > x > 2, which no number satisfies.

  Wrong working

  Solution: 2 < x < −3.

  Right working

  Solution: x < −3 or x > 2, i.e. (−∞, −3) ∪ (2, ∞).

  ✓Do this: Use a chain only when the set is a single interval.

  Usually costs: 1 mark.

• ✗Mistake: Dismissing a perfect square

  Students say x² − 6x + 9 > 0 is true for all x.

  Wrong working

  x² − 6x + 9 > 0 ⇒ x ∈ ℝ.

  Right working

  (x − 3)² > 0 ⇒ x ∈ ℝ, x ≠ 3.

  ✓Do this: Check what happens at the repeated root.

  Usually costs: 1 mark.

Practice questions on Linear and Quadratic Inequalities

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

1. Q1 · 1 mark · MCQExam level

   If x is a natural number, the solution set of 2x − 5 ≤ 1 is:
   (a) {1, 2}
   (b) {1, 2, 3}
   (c) {0, 1, 2, 3}
   (d) {3, 4, 5, …}

   Hint

   x ≤ 3.

   Answer

   (b)

2. Q2 · 1 mark · MCQExam level

   The solution of (x − 1)(x − 4) < 0 is:
   (a) x < 1 or x > 4
   (b) 1 < x < 4
   (c) 1 ≤ x ≤ 4
   (d) x > 4

   Hint

   Between the roots.

   Answer

   (b)

3. Q3 · 1 mark · Very short answerExam level

   Write −2 < x ≤ 5 in interval notation.

   Hint

   Round bracket for <.

   Answer

   (−2, 5]

4. Q4 · 1 mark · Very short answerExam level

   Solve −4x ≥ 12.

   Hint

   Flip the sign.

   Answer

   x ≤ −3

5. Q5 · 2 marks · Short answerExam level

   Solve x/3 − (x − 1)/4 > 1.

   Hint

   Multiply by 12.

   Answer

   4x − 3x + 3 > 12 ⇒ x > 9

6. Q6 · 2 marks · Short answerHarder exam level

   Solve x² − 9 ≤ 0.

   Hint

   (x − 3)(x + 3).

   Answer

   −3 ≤ x ≤ 3

7. Q7 · 2 marks · Short answerHarder exam level

   Solve 3x² + x − 2 > 0.

   Hint

   (3x − 2)(x + 1).

   Answer

   x < −1 or x > 2/3

8. Q8 · 3 marks · Short answerHarder exam level

   Solve 2 ≤ 3x − 4 < 11 and represent the solution on a number line.

   Hint

   Add 4 to all parts.

   Answer

   6 ≤ 3x < 15 ⇒ 2 ≤ x < 5: filled dot at 2, open dot at 5, shading between.

9. Q9 · 3 marks · Short answerHarder exam level

   Solve 1/(x − 2) < 1/x.

   Hint

   Subtract and combine.

   Answer

   1/(x − 2) − 1/x = 2/(x(x − 2)) < 0 ⇒ x(x − 2) < 0 ⇒ 0 < x < 2.

10. Q10 · 2 marks · Short answerHarder exam level

    Solve (i) x² + 6x + 9 > 0 (ii) x² + 6x + 9 ≤ 0.

    Hint

    (x + 3)².

    Answer

    (i) all real x except −3 (ii) x = −3 only

11. Q11 · 3 marks · Short answerHarder exam level

    Solve x/(x − 1) > 2.

    Hint

    One side zero.

    Answer

    (x − 2x + 2)/(x − 1) > 0 ⇒ (2 − x)/(x − 1) > 0 ⇒ (x − 2)/(x − 1) < 0 ⇒ 1 < x < 2.

12. Q12 · 5 marks · Long answerHarder exam level

    (a) Solve the system 2(x − 1) < x + 5 and 3x + 2 > 2x − 1. [2]
    (b) Solve x² − 2x − 15 ≤ 0. [2]
    (c) Find the values of x satisfying both (a) and (b). [1]

    Hint

    Intersect on one number line.

    Answer

    (a) x < 7 and x > −3 ⇒ −3 < x < 7. (b) (x − 5)(x + 3) ≤ 0 ⇒ −3 ≤ x ≤ 5. (c) −3 < x ≤ 5.

13. Q13 · 3 marks · Short answerHarder exam level

    Solve (x − 2)²(x − 1)(x − 4) < 0.

    Hint

    No sign change at 2.

    Answer

    (x − 1)(x − 4) < 0 gives 1 < x < 4, and x = 2 makes the product zero, so 1 < x < 4, x ≠ 2.

14. Q14 · 3 marks · Short answerHOTS / competency

    Find all real m for which mx² − 4x + m < 0 for every real x.

    Hint

    The parabola must open downwards and miss the axis.

    Answer

    Need m < 0 and D = 16 − 4m² < 0, i.e. m² > 4. With m < 0: m < −2.

15. Q15 · 3 marks · Short answerHOTS / competency

    Solve (x² + 2)/(x² − 1) ≤ 3.

    Hint

    Get (2x² − 5)/(x² − 1) ≥ 0.

    Answer

    (x² + 2 − 3x² + 3)/(x² − 1) ≤ 0 ⇒ (2x² − 5)/(x² − 1) ≥ 0. Critical points ±√(5/2), ±1. Solution: x ≤ −√(5/2) or −1 < x < 1 or x ≥ √(5/2).

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1. 1.What happens when you divide an inequality by a negative number?Show answer

   The inequality sign reverses.

2. 2.Filled or open dot for x ≤ 4?Show answer

   Filled dot at 4.

3. 3.(x − α)(x − β) < 0 with α < β?Show answer

   α < x < β.

4. 4.(x − α)(x − β) > 0 with α < β?Show answer

   x < α or x > β.

5. 5.Solution of (x − 3)² ≤ 0?Show answer

   x = 3 only.

6. 6.Condition for ax² + bx + c > 0 for all real x?Show answer

   a > 0 and D < 0.

7. 7.Why not cross-multiply in f(x)/g(x) ≤ a?Show answer

   g(x) may be negative, which would reverse the sign.

8. 8.Are zeros of the denominator ever included?Show answer

   Never.

Quick revision checklist

The night before the exam: can you do each of these without looking?

• ☐Can you solve a linear inequality with fractions without sign errors?
• ☐Can you draw the solution on a number line with correct dots?
• ☐Can you solve a system of two or three linear inequalities?
• ☐Can you solve ax² + bx + c ≥ 0 by the method of intervals?
• ☐Can you handle x² − 6x + 9 ≥ 0 and x² − 6x + 9 < 0 correctly?
• ☐Can you solve f(x)/g(x) ≤ a without cross-multiplying?
• ☐Can you deal with a squared factor in a sign chart?
• ☐Can you find k so that a quadratic is positive for all real x?

Linear and Quadratic Inequalities: frequently asked questions