Inequalities – Complete Notes for Competitive Exams

1. Introduction

Inequalities are mathematical statements that compare two expressions using symbols like:$$<, >, \le, \ge, \neq$$<,>,≤,≥,=

They are a frequent topic in Quantitative Aptitude sections of SSC, Banking, Railways, and other competitive exams.

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2. Types of Inequalities

1. Linear Inequalities: e.g., $2x + 3 > 7$2x+3>7
2. Quadratic Inequalities: e.g., $x^2 – 5x + 6 \le 0$x2−5x+6≤0
3. Rational Inequalities: e.g., $\frac{x-1}{x+2} \ge 0$x+2x−1​≥0
4. Absolute Value Inequalities: e.g., $|x-3| < 5$∣x−3∣<5

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3. Basic Properties of Inequalities

• Addition/Subtraction: If $a > b$a>b, then $a + c > b + c$a+c>b+c
• Multiplication/Division:
  • If $c > 0$c>0, $a > b \implies ac > bc$a>b⟹ac>bc
  • If $c < 0$c<0, $a > b \implies ac < bc$a>b⟹ac<bc
• Transitive Property: If $a > b$a>b and $b > c$b>c, then $a > c$a>c
• Reversal Rule: Multiply or divide by a negative number → inequality reverses

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4. Solving Linear Inequalities

1. Isolate the variable on one side
2. Reverse inequality sign if multiplied/divided by negative
3. Express solution in interval or set notation

Example: Solve $3x – 5 > 1$3x−5>1$$3x > 6 \implies x > 2$$3x>6⟹x>2

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5. Solving Quadratic Inequalities

1. Factorize quadratic expression
2. Find roots $x_1, x_2$x1​,x2​
3. Determine intervals using sign chart

Example: Solve $x^2 – 5x + 6 < 0$x2−5x+6<0$$(x-2)(x-3) < 0 \implies 2 < x < 3$$(x−2)(x−3)<0⟹2<x<3

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6. Absolute Value Inequalities

• $|x – a| < b \implies a-b < x < a+b$∣x−a∣<b⟹a−b<x<a+b
• $|x – a| > b \implies x < a-b \text{ or } x > a+b$∣x−a∣>b⟹x<a−b or x>a+b

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7. Important Tips

• Always check the sign of coefficients
• Use interval method for quadratics and rationals
• Be careful with negative multipliers
• Practice word problems involving inequalities

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Top 25 Practice Questions – Inequalities

Linear Inequalities

Q1. Solve $2x + 5 > 9$2x+5>9
Q2. Solve $3x – 7 \le 2$3x−7≤2
Q3. Solve $-4x + 5 > 1$−4x+5>1
Q4. Solve $\frac{x}{2} + 3 \le 7$2x​+3≤7
Q5. Solve $5 – 2x \ge 1$5−2x≥1
Q6. Solve $7x + 3 < 4x + 12$7x+3<4x+12
Q7. Solve $-3x + 2 \le 11$−3x+2≤11
Q8. Solve $4x – 5 > 3x + 2$4x−5>3x+2
Q9. Solve $2x + 1 \le 3x – 4$2x+1≤3x−4
Q10. Solve $-5x + 7 < 2x – 8$−5x+7<2x−8

Quadratic & Absolute Value Inequalities

Q11. Solve $x^2 – 5x + 6 \le 0$x2−5x+6≤0
Q12. Solve $x^2 + x – 6 > 0$x2+x−6>0
Q13. Solve $x^2 – 9 < 0$x2−9<0
Q14. Solve $(x-3)(x+2) \ge 0$(x−3)(x+2)≥0
Q15. Solve $|x-4| < 3$∣x−4∣<3
Q16. Solve $|x+2| \ge 5$∣x+2∣≥5
Q17. Solve $x^2 – 4x – 5 \le 0$x2−4x−5≤0
Q18. Solve $x^2 + 2x – 8 > 0$x2+2x−8>0
Q19. Solve $|2x-3| < 7$∣2x−3∣<7
Q20. Solve $|x-1| > 4$∣x−1∣>4
Q21. Solve $x^2 – x – 6 \ge 0$x2−x−6≥0
Q22. Solve $(x-1)(x-4) < 0$(x−1)(x−4)<0
Q23. Solve $|3x+2| \le 8$∣3x+2∣≤8
Q24. Solve $x^2 – 7x + 10 > 0$x2−7x+10>0
Q25. Solve $|x-5| \ge 2$∣x−5∣≥2

Answer

Answers – Inequalities

Linear Inequalities Answers

Q1. $x > 2$x>2
Q2. $x \le 3$x≤3
Q3. $x < 1$x<1
Q4. $x \le 8$x≤8
Q5. $x \le 2$x≤2
Q6. $x < 3$x<3
Q7. $x \ge -3$x≥−3
Q8. $x > 7$x>7
Q9. $x \ge 5$x≥5
Q10. $x > 1$x>1

Quadratic & Absolute Value Inequalities Answers

Q11. $2 \le x \le 3$2≤x≤3
Q12. $x < -3$x<−3 or $x > 2$x>2
Q13. $-3 < x < 3$−3<x<3
Q14. $x \le -2$x≤−2 or $x \ge 3$x≥3
Q15. $1 < x < 7$1<x<7
Q16. $x \le -7$x≤−7 or $x \ge 3$x≥3
Q17. $-1 \le x \le 5$−1≤x≤5
Q18. $x < -4$x<−4 or $x > 2$x>2
Q19. $-2 < x < 5$−2<x<5
Q20. $x < -3$x<−3 or $x > 5$x>5
Q21. $x \le -2$x≤−2 or $x \ge 3$x≥3
Q22. $1 < x < 4$1<x<4
Q23. $-10/3 \le x \le 2$−10/3≤x≤2
Q24. $x < 2$x<2 or $x > 5$x>5
Q25. $x \le 3$x≤3 or $x \ge 7$x≥7