Linear Inequalities — Class 11 Maths Notes

Topics covered in Linear Inequalities

1. 1.Introduction of Linear Inequalities

   A linear inequality is similar to a linear equation but instead of an equal sign, it uses inequality symbols. These symbols are:
   

   • > (greater than)
   • < (less than)
   • ≥ (greater than or equal to)
   • ≤ (less than or equal to)

   A linear inequality looks like this: 𝑎𝑥+𝑏>𝑐ax+b>c or 𝑎𝑥+𝑏≤𝑐ax+b≤c, where 𝑎a, 𝑏b, and 𝑐c are constants.

   Solving Linear Inequalities

   1. Isolate the variable on one side: Just like solving an equation, you perform operations to get the variable (e.g., 𝑥x) by itself on one side of the inequality.

   2. Perform operations: Whatever operation you do to one side, you must do to the other side to maintain the inequality.

   3. Flip the inequality sign when multiplying or dividing by a negative number: This is crucial to maintain the correct relationship between the numbers.

   Example:

   Solve the inequality: 2𝑥−3<72x−3<7

   1. Add 3 to both sides: 2𝑥−3+3<7+32x−3+3<7+3 2𝑥<102x<10

   2. Divide by 2: 𝑥<5x<5

   So, 𝑥x can be any value less than 5.

   Graphing Linear Inequalities

   To visualize linear inequalities, you can graph them on a number line or coordinate plane. For a simple inequality like 𝑥<5x<5:

   1. Draw a number line.
   2. Mark 5 with an open circle (because 5 is not included).
   3. Shade everything to the left of 5 (because 𝑥x can be any number less than 5).

   For inequalities with two variables (e.g., 𝑦≤2𝑥+3y≤2x+3):

   1. First, draw the boundary line 𝑦=2𝑥+3y=2x+3. Use a dashed line for "<" or ">" and a solid line for "≤" or "≥".
   2. Shade the region that satisfies the inequality.

   Real-Life Example

   Imagine you have a monthly allowance of ₹500. You want to save at least ₹200 each month. This situation can be represented by the inequality:

   𝑥≤500−200x≤500−200

   Here, 𝑥

   x represents your spending. The inequality means you can spend up to ₹300 to save at least ₹200. Applications in Real Life and Careers

   Linear inequalities are used in various fields:

   • Business: To determine profit margins and cost constraints.
   • Economics: For budget limitations and resource allocation.
   • Engineering: To ensure safety limits and specifications.
   • Daily life: Managing budgets, comparing prices, and making decisions based on constraints.

2. 2.Inequalities

   What Are Inequalities?

   Inequalities are mathematical expressions that show the relationship between two values, where one value is not equal to the other. Instead, they express whether one value is greater than, less than, greater than or equal to, or less than or equal to another value. The symbols used to denote inequalities are:

   • > (greater than)
   • < (less than)
   • ≥ (greater than or equal to)
   • ≤ (less than or equal to)
   • ≠ (not equal to)

   Basic Types of Inequalities

   1. Linear Inequalities: These are inequalities that involve linear expressions. For example:

      • 2𝑥+3>72x+3>7
      • −𝑥+4≤10−x+4≤10

   2. Quadratic Inequalities: These involve quadratic expressions. For example:

      • 𝑥2−4𝑥+3≥0x2−4x+3≥0
      • 𝑥2+𝑥−6<0x2+x−6<0

   3. Polynomial Inequalities: These involve polynomial expressions of degree higher than 2. For example:

      • 𝑥3−𝑥2−𝑥+1>0x3−x2−x+1>0

   4. Rational Inequalities: These involve rational expressions (ratios of polynomials). For example:

      • 𝑥+1𝑥−2<3x−2x+1​<3

   Solving Inequalities

   Linear Inequalities

   To solve linear inequalities, follow these steps:

   1. Isolate the variable: Get the variable on one side of the inequality.
   2. Simplify: Perform basic algebraic operations to simplify the inequality.
   3. Reverse the inequality sign when multiplying or dividing by a negative number: This is crucial to maintaining the correct relationship.

   Example:

   Solve 2𝑥−3<72x−3<7:

   1. Add 3 to both sides: 2𝑥−3+3<7+32x−3+3<7+3 2𝑥<102x<10

   2. Divide by 2: 𝑥<5x<5

   So, the solution is 𝑥<5x<5.

   Quadratic Inequalities

   To solve quadratic inequalities, follow these steps:

   1. Rewrite the inequality as an equation: Set the inequality to zero.
   2. Factor or use the quadratic formula: Find the roots of the equation.
   3. Determine the intervals: Use the roots to divide the number line into intervals.
   4. Test the intervals: Check each interval to see where the inequality holds true.

   Example:

   Solve 𝑥2−4𝑥+3≥0x2−4x+3≥0:

   1. Rewrite as an equation: 𝑥2−4𝑥+3=0x2−4x+3=0
   2. Factor: (𝑥−1)(𝑥−3)=0(x−1)(x−3)=0
   3. Find roots: 𝑥=1x=1, 𝑥=3x=3
   4. Determine intervals: (−∞,1)(−∞,1), (1,3)(1,3), (3,∞)(3,∞)
   5. Test intervals:
      • For 𝑥∈(−∞,1)x∈(−∞,1): Choose 𝑥=0x=0, 02−4(0)+3=3≥002−4(0)+3=3≥0 (True)
      • For 𝑥∈(1,3)x∈(1,3): Choose 𝑥=2x=2, 22−4(2)+3=−1≥022−4(2)+3=−1≥0 (False)
      • For 𝑥∈(3,∞)x∈(3,∞): Choose 𝑥=4x=4, 42−4(4)+3=3≥042−4(4)+3=3≥0 (True)

   So, the solution is 𝑥∈(−∞,1]∪[3,∞)x∈(−∞,1]∪[3,∞).

   Rational Inequalities

   To solve rational inequalities, follow these steps:

   1. Rewrite the inequality as an equation: Set the inequality to zero.
   2. Find the critical points: Determine where the numerator and denominator are zero.
   3. Determine the intervals: Use the critical points to divide the number line into intervals.
   4. Test the intervals: Check each interval to see where the inequality holds true.

   Example:

   Solve 𝑥+1𝑥−2<3x−2x+1​<3:

   1. Rewrite as an equation: 𝑥+1𝑥−2−3<0x−2x+1​−3<0 𝑥+1−3(𝑥−2)𝑥−2<0x−2x+1−3(x−2)​<0 𝑥+1−3𝑥+6𝑥−2<0x−2x+1−3x+6​<0 −2𝑥+7𝑥−2<0x−2−2x+7​<0

   2. Find critical points: −2𝑥+7=0−2x+7=0 gives 𝑥=72x=27​, and 𝑥−2=0x−2=0 gives 𝑥=2x=2.

   3. Determine intervals: (−∞,2)(−∞,2), (2,72)(2,27​), (72,∞)(27​,∞)

   4. Test intervals:

      • For 𝑥∈(−∞,2)x∈(−∞,2): Choose 𝑥=0x=0, −2(0)+70−2=−72<00−2−2(0)+7​=−27​<0 (False)
      • For 𝑥∈(2,72)x∈(2,27​): Choose 𝑥=3x=3, −2(3)+73−2=11=1<03−2−2(3)+7​=11​=1<0 (False)
      • For 𝑥∈(72,∞)x∈(27​,∞): Choose 𝑥=4x=4, −2(4)+74−2=−12<04−2−2(4)+7​=−21​<0 (True)

   So, the solution is 𝑥∈(72,∞)x∈(27​,∞).

   Graphing Inequalities

   On a Number Line

   For inequalities involving a single variable:

   1. Plot critical points: Use open circles for strict inequalities (>, <) and closed circles for inclusive inequalities (≥, ≤).
   2. Shade the regions: Shade the portion of the number line that satisfies the inequality.

   Example: 𝑥>3x>3

   • Plot an open circle at 3.
   • Shade everything to the right of 3.

   In a Coordinate Plane

   For inequalities involving two variables:

   1. Graph the boundary line: Use a dashed line for strict inequalities (>, <) and a solid line for inclusive inequalities (≥, ≤).
   2. Shade the appropriate region: Shade above the line for > or ≥, and below the line for < or ≤.

   Example: 𝑦≤2𝑥+3y≤2x+3

   • Draw the line 𝑦=2𝑥+3y=2x+3 using a solid line.
   • Shade the region below the line.

   Applications in Real Life and Careers

   Inequalities are used in various real-life scenarios and careers, such as:

   • Budgeting: Ensuring expenses do not exceed income.
   • Engineering: Defining safe operating conditions.
   • Business: Determining profit margins and cost constraints.
   • Economics: Allocating resources efficiently.
   • Data Science: Setting thresholds for data analysis.

3. 3.Algebraic Solutions of Linear Inequalities in One Variable and their Graphical Representation.

   Understanding Linear Inequalities in One Variable

   A linear inequality in one variable is an inequality that can be written in one of the following forms: 𝑎𝑥+𝑏>𝑐ax+b>c 𝑎𝑥+𝑏<𝑐ax+b<c 𝑎𝑥+𝑏≥𝑐ax+b≥c 𝑎𝑥+𝑏≤𝑐ax+b≤c

   where 𝑎a, 𝑏b, and 𝑐c are constants, and 𝑥x is the variable.

   Solving Linear Inequalities Algebraically

   To solve a linear inequality algebraically, follow these steps:

   1. Isolate the Variable: Perform operations to get the variable 𝑥x on one side of the inequality.
   2. Simplify the Inequality: Simplify both sides of the inequality.
   3. Reverse the Inequality Sign if Necessary: When you multiply or divide both sides by a negative number, you must reverse the inequality sign.

   Example:

   Solve 2𝑥−3<72x−3<7:

   1. Add 3 to both sides: 2𝑥−3+3<7+32x−3+3<7+3 2𝑥<102x<10

   2. Divide both sides by 2: 𝑥<5x<5

   So, the solution is 𝑥<5x<5.

   Graphical Representation of Linear Inequalities in One Variable

   Graphing the solution of a linear inequality involves showing the range of values that satisfy the inequality on a number line.

   Steps to Graph Linear Inequalities

   1. Identify the Boundary Point: The boundary point is the value of 𝑥x that makes the inequality an equation (e.g., 𝑥=5x=5 in 𝑥<5x<5).
   2. Use Open or Closed Circles:
      • Use an open circle for << or >> to indicate the boundary point is not included.
      • Use a closed circle for ≤≤ or ≥≥ to indicate the boundary point is included.
   3. Shade the Appropriate Region: Shade the number line to show the range of values that satisfy the inequality.

   Detailed Example

   Inequality: 𝑥−2≥3x−2≥3

   Step-by-Step Solution:

   1. Add 2 to both sides: 𝑥−2+2≥3+2x−2+2≥3+2 𝑥≥5x≥5

   2. Boundary Point: 𝑥=5x=5

   3. Closed Circle: Place a closed circle at 5 to indicate 𝑥=5x=5 is included.

   4. Shade the Region: Shade to the right of 5 to indicate all values greater than or equal to 5.

   Graphical Representation:

   1. Draw a number line: Mark key points on the number line.
   2. Locate the boundary point: In this case, 𝑥=5x=5.
   3. Place a closed circle at 5: Because the inequality is ≥≥, include the boundary point.
   4. Shade to the right of 5: This shows that all values of 𝑥x greater than or equal to 5 are solutions.

   Another Example:

   Inequality: 𝑥+4<2x+4<2

   Step-by-Step Solution:

   1. Subtract 4 from both sides: 𝑥+4−4<2−4x+4−4<2−4 𝑥<−2x<−2

   2. Boundary Point: 𝑥=−2x=−2

   3. Open Circle: Place an open circle at -2 to indicate 𝑥=−2x=−2 is not included.

   4. Shade the Region: Shade to the left of -2 to indicate all values less than -2.

   Graphical Representation:

   1. Draw a number line: Mark key points on the number line.
   2. Locate the boundary point: In this case, 𝑥=−2x=−2.
   3. Place an open circle at -2: Because the inequality is <<, do not include the boundary point.
   4. Shade to the left of -2: This shows that all values of 𝑥x less than -2 are solutions.

   Real-Life Applications of Linear Inequalities

   Linear inequalities are used in various real-life situations, such as:

   • Budgeting: Ensuring expenses do not exceed a certain limit.
   • Age Restrictions: Determining minimum age requirements for activities.
   • Stock Management: Keeping inventory levels above a certain threshold.

   Careers Using Linear Inequalities

   • Economics: Budgeting and resource allocation.
   • Engineering: Design specifications and safety limits.
   • Business: Profit and cost management.

4. 4.Exercise Question

   1. Solve 24𝑥<10024x<100:

   (i) 𝑥x is a natural number:

   Natural numbers are {1,2,3,…}{1,2,3,…}.

   24𝑥<10024x<100 𝑥<10024x<24100​ 𝑥<4.1667x<4.1667

   Since 𝑥x is a natural number: 𝑥∈{1,2,3,4}x∈{1,2,3,4}

   (ii) 𝑥x is an integer:

   Integers include both positive and negative whole numbers {…,−2,−1,0,1,2,…}{…,−2,−1,0,1,2,…}.

   24𝑥<10024x<100 𝑥<10024x<24100​ 𝑥<4.1667x<4.1667

   Since 𝑥x is an integer: 𝑥∈{…,−2,−1,0,1,2,3,4}x∈{…,−2,−1,0,1,2,3,4}

   2. Solve −12𝑥>30−12x>30:

   (i) 𝑥x is a natural number:

   −12𝑥>30−12x>30 𝑥<30−12x<−1230​ 𝑥<−2.5x<−2.5

   Since natural numbers are positive, there are no natural numbers that satisfy this inequality.

   (ii) 𝑥x is an integer:

   −12𝑥>30−12x>30 𝑥<30−12x<−1230​ 𝑥<−2.5x<−2.5

   Since 𝑥x is an integer: 𝑥∈{…,−4,−3}x∈{…,−4,−3}

   3. Solve 5𝑥−3<75x−3<7:

   (i) 𝑥x is an integer:

   5𝑥−3<75x−3<7 5𝑥<105x<10 𝑥<2x<2

   Since 𝑥x is an integer: 𝑥∈{…,−2,−1,0,1}x∈{…,−2,−1,0,1}

   (ii) 𝑥x is a real number:

   Real numbers include all rational and irrational numbers.

   5𝑥−3<75x−3<7 5𝑥<105x<10 𝑥<2x<2

   All real numbers less than 2 satisfy this inequality: 𝑥∈(−∞,2)x∈(−∞,2)

   4. Solve 3𝑥+8>23x+8>2:

   (i) 𝑥x is an integer:

   3𝑥+8>23x+8>2 3𝑥>−63x>−6 𝑥>−2x>−2

   Since 𝑥x is an integer: 𝑥∈{…,−1,0,1,2,…}x∈{…,−1,0,1,2,…}

   (ii) 𝑥x is a real number:

   3𝑥+8>23x+8>2 3𝑥>−63x>−6 𝑥>−2x>−2

   All real numbers greater than -2 satisfy this inequality: 𝑥∈(−2,∞)x∈(−2,∞)

   5. Solve 4𝑥+3<5𝑥+74x+3<5x+7:

   Subtract 4𝑥4x from both sides: 3<𝑥+73<x+7

   Subtract 7 from both sides: −4<𝑥−4<x 𝑥>−4x>−4

   So, the solution is: 𝑥>−4x>−4

   6. Solve 3𝑥−7>5𝑥−13x−7>5x−1:

   Subtract 3𝑥3x from both sides: −7>2𝑥−1−7>2x−1

   Add 1 to both sides: −6>2𝑥−6>2x

   Divide both sides by 2: −3>𝑥−3>x 𝑥<−3x<−3

   So, the solution is: 𝑥<−3x<−3

   7. Solve 3(𝑥−1)≤2(𝑥−3)3(x−1)≤2(x−3):

   Expand both sides: 3𝑥−3≤2𝑥−63x−3≤2x−6

   Subtract 2𝑥2x from both sides: 𝑥−3≤−6x−3≤−6

   Add 3 to both sides: 𝑥≤−3x≤−3

   So, the solution is: 𝑥≤−3x≤−3

   8. Solve 3(2−𝑥)≥2(1−𝑥)3(2−x)≥2(1−x):

   Expand both sides: 6−3𝑥≥2−2𝑥6−3x≥2−2x

   Add 3𝑥3x to both sides: 6≥2+𝑥6≥2+x

   Subtract 2 from both sides: 4≥𝑥4≥x 𝑥≤4x≤4

   So, the solution is: 𝑥≤4x≤4

   9. Solve 𝑥+𝑥2−𝑥3<11x+2x​−3x​<11:

   Find a common denominator for the fractions: 𝑥+3𝑥6−2𝑥6<11x+63x​−62x​<11

   Combine the fractions: 𝑥+𝑥6<11x+6x​<11

   Combine the terms: 6𝑥+𝑥6<1166x+x​<11 7𝑥6<1167x​<11

   Multiply both sides by 6: 7𝑥<667x<66

   Divide both sides by 7: 𝑥<667x<766​ 𝑥<9.4286x<9.4286

   So, the solution is: 𝑥<9.4286x<9.4286

   10. Solve 𝑥3−𝑥2≥13x​−2x​≥1:

   Find a common denominator for the fractions: 2𝑥6−3𝑥6≥162x​−63x​≥1

   Combine the fractions: 2𝑥−3𝑥6≥162x−3x​≥1 −𝑥6≥16−x​≥1

   Multiply both sides by -6 (note that this reverses the inequality): −𝑥≥6−x≥6 𝑥≤−6x≤−6

   So, the solution is: 𝑥≤−6x≤−6

5. 5.Exercise Question

   1. Solve 2≤3𝑥−4≤52≤3x−4≤5:

   Break it into two parts and solve each part:

   Part 1: 2≤3𝑥−42≤3x−4

   Add 4 to both sides: 2+4≤3𝑥2+4≤3x 6≤3𝑥6≤3x

   Divide both sides by 3: 2≤𝑥2≤x

   Part 2: 3𝑥−4≤53x−4≤5

   Add 4 to both sides: 3𝑥≤5+43x≤5+4 3𝑥≤93x≤9

   Divide both sides by 3: 𝑥≤3x≤3

   Combine the results: 2≤𝑥≤32≤x≤3

   2. Solve 6≤3(2𝑥−4)≤126≤3(2x−4)≤12:

   Break it into two parts and solve each part:

   Part 1: 6≤3(2𝑥−4)6≤3(2x−4)

   Divide both sides by 3: 2≤2𝑥−42≤2x−4

   Add 4 to both sides: 2+4≤2𝑥2+4≤2x 6≤2𝑥6≤2x

   Divide both sides by 2: 3≤𝑥3≤x

   Part 2: 3(2𝑥−4)≤123(2x−4)≤12

   Divide both sides by 3: 2𝑥−4≤42x−4≤4

   Add 4 to both sides: 2𝑥≤4+42x≤4+4 2𝑥≤82x≤8

   Divide both sides by 2: 𝑥≤4x≤4

   Combine the results: 3≤𝑥≤43≤x≤4

   3. Solve −3≤4−7𝑥2≤18−3≤4−27x​≤18:

   Break it into two parts and solve each part:

   Part 1: −3≤4−7𝑥2−3≤4−27x​

   Subtract 4 from both sides: −3−4≤−7𝑥2−3−4≤−27x​ −7≤−7𝑥2−7≤−27x​

   Multiply both sides by -2/7 (and reverse the inequality): 147≥𝑥714​≥x 2≥𝑥2≥x 𝑥≤2x≤2

   Part 2: 4−7𝑥2≤184−27x​≤18

   Subtract 4 from both sides: −7𝑥2≤18−4−27x​≤18−4 −7𝑥2≤14−27x​≤14

   Multiply both sides by -2/7 (and reverse the inequality): 𝑥≥−4x≥−4

   Combine the results: −4≤𝑥≤2−4≤x≤2

   4. Solve −15≤3(𝑥−2)5≤0−15≤53(x−2)​≤0:

   Break it into two parts and solve each part:

   Part 1: −15≤3(𝑥−2)5−15≤53(x−2)​

   Multiply both sides by 5: −75≤3(𝑥−2)−75≤3(x−2)

   Divide both sides by 3: −25≤𝑥−2−25≤x−2

   Add 2 to both sides: −23≤𝑥−23≤x

   Part 2: 3(𝑥−2)5≤053(x−2)​≤0

   Multiply both sides by 5: 3(𝑥−2)≤03(x−2)≤0

   Divide both sides by 3: 𝑥−2≤0x−2≤0

   Add 2 to both sides: 𝑥≤2x≤2

   Combine the results: −23≤𝑥≤2−23≤x≤2

   5. Solve −12<4−3𝑥5≤2−12<4−53x​≤2:

   Break it into two parts and solve each part:

   Part 1: −12<4−3𝑥5−12<4−53x​

   Subtract 4 from both sides: −16<−3𝑥5−16<−53x​

   Multiply both sides by -5/3 (and reverse the inequality): 𝑥<803x<380​ 𝑥<26.67x<26.67

   Part 2: 4−3𝑥5≤24−53x​≤2

   Subtract 4 from both sides: −3𝑥5≤−2−53x​≤−2

   Multiply both sides by -5/3 (and reverse the inequality): 𝑥≥103x≥310​ 𝑥≥3.33x≥3.33

   Combine the results: 3.33≤𝑥<26.673.33≤x<26.67

   6. Solve 7≤3𝑥+112≤117≤23x+11​≤11:

   Break it into two parts and solve each part:

   Part 1: 7≤3𝑥+1127≤23x+11​

   Multiply both sides by 2: 14≤3𝑥+1114≤3x+11

   Subtract 11 from both sides: 3≤3𝑥3≤3x

   Divide both sides by 3: 1≤𝑥1≤x

   Part 2: 3𝑥+112≤1123x+11​≤11

   Multiply both sides by 2: 3𝑥+11≤223x+11≤22

   Subtract 11 from both sides: 3𝑥≤113x≤11

   Divide both sides by 3: 𝑥≤113x≤311​ 𝑥≤3.67x≤3.67

   Combine the results: 1≤𝑥≤3.671≤x≤3.67