Linear Inequalities Ex 6.1: Master CBSE Class 11 Maths

What are Linear Inequalities?

A linear inequality is a mathematical statement that compares two expressions using an inequality symbol: < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). Unlike linear equations, which have a single solution or no solution, linear inequalities often have an infinite set of solutions that can be represented on a number line. In Class 11, we primarily deal with linear inequalities involving one variable. The fundamental rules for manipulating inequalities are similar to those for equations, with one crucial difference: multiplying or dividing both sides by a negative number reverses the direction of the inequality sign. This distinction is vital for finding the correct solution set and is a common area for mistakes if not handled carefully.

Step-by-Step Guide to Solving Linear Inequalities

1. Simplify Both Sides — If necessary, simplify each side of the inequality by distributing, combining like terms, or performing any indicated operations. The goal is to get the inequality into a simpler form like ax + b < cx + d.
2. Isolate the Variable Terms — Move all terms containing the variable to one side of the inequality (usually the left) and all constant terms to the other side. Do this by adding or subtracting the same quantity from both sides. Remember, adding or subtracting does not change the direction of the inequality sign.
3. Isolate the Variable — Divide or multiply both sides by the coefficient of the variable to solve for the variable. This is the most critical step: if you multiply or divide by a negative number, you must reverse the direction of the inequality sign. If you multiply or divide by a positive number, the sign remains unchanged.
4. Express the Solution Set — Write down the solution set using appropriate notation, such as interval notation or set-builder notation, and, if required, represent it on a number line. For example, x > 5 can be written as (5, ∞).

Representing Solutions on a Number Line

Once you've solved a linear inequality, it's often helpful to visualise its solution set on a number line. This graphical representation clarifies the range of values that satisfy the inequality. For inequalities involving < or >, we use an open circle (or a parenthesis ( or )) at the boundary point, indicating that the point itself is not included in the solution set. For inequalities with ≤ or ≥, we use a closed circle (or a square bracket [ or ]) at the boundary point, meaning the point is included. The part of the number line that satisfies the inequality is then shaded. For example, x > 3 would be an open circle at 3, with shading to the right, whereas x ≤ -2 would be a closed circle at -2, with shading to the left.

Solved Examples from Ex 6.1

• Example 1: Solve 24x < 100 when (i) x is a natural number (ii) x is an integer. Solution: Step 1: Isolate x in the inequality. 24x < 100 Divide both sides by 24 (a positive number, so inequality sign remains the same): x < 100 / 24 x < 25 / 6 x < 4.166... Step 2: Consider the condition for x. (i) If x is a natural number, the natural numbers less than 4.166... are 1, 2, 3, 4. Solution set: {1, 2, 3, 4} (ii) If x is an integer, the integers less than 4.166... are ..., -2, -1, 0, 1, 2, 3, 4. Solution set: {..., -2, -1, 0, 1, 2, 3, 4}
• Example 2: Solve 5x - 3 < 3x + 7 for real x. Solution: Step 1: Simplify and collect variable terms on one side. 5x - 3 < 3x + 7 Subtract 3x from both sides: 5x - 3x - 3 < 7 2x - 3 < 7 Step 2: Collect constant terms on the other side. Add 3 to both sides: 2x < 7 + 3 2x < 10 Step 3: Isolate x. Divide both sides by 2 (a positive number, sign remains the same): x < 10 / 2 x < 5 Step 4: Express the solution set for real x. The solution is all real numbers less than 5. Solution set: (-∞, 5)
• Example 3: Solve (3x - 4) / 2 ≥ (x + 1) / 4 - 1 for real x. Solution: Step 1: Clear the fractions by multiplying by the LCM of denominators. The denominators are 2 and 4. LCM is 4. Multiply the entire inequality by 4: 4 [(3x - 4) / 2] ≥ 4 [(x + 1) / 4] - 4 * 1 2(3x - 4) ≥ (x + 1) - 4 6x - 8 ≥ x + 1 - 4 6x - 8 ≥ x - 3 Step 2: Isolate variable terms. Subtract x from both sides: 6x - x - 8 ≥ -3 5x - 8 ≥ -3 Step 3: Isolate constant terms. Add 8 to both sides: 5x ≥ -3 + 8 5x ≥ 5 Step 4: Isolate x. Divide both sides by 5 (a positive number, sign remains the same): x ≥ 5 / 5 x ≥ 1 Step 5: Express the solution set for real x. The solution is all real numbers greater than or equal to 1. Solution set: [1, ∞)

Exam Tips for Linear Inequalities

When solving linear inequalities, especially in exams, pay close attention to these crucial points to avoid common mistakes:

1. Reversing the Inequality Sign: This is the most frequent error. Always remember to reverse the inequality sign (e.g., < becomes >, ≤ becomes ≥) when you multiply or divide both sides of the inequality by a negative number. If you forget this, your entire solution will be incorrect.
2. Solution Set Domain: Carefully read whether 'x' belongs to natural numbers, integers, or real numbers. The solution set will vary significantly based on this domain. For instance, x < 3 means {1, 2} for natural numbers, {..., 0, 1, 2} for integers, and (-∞, 3) for real numbers.
3. Interval Notation: Be precise with interval notation. Use parentheses ( ) for strict inequalities (<, >) and square brackets [ ] for inclusive inequalities (≤, ≥). Infinity (∞ or -∞) always uses parentheses.
4. Combining Terms: Ensure you correctly combine like terms and distribute properly before isolating the variable. Algebraic errors here can lead to wrong results.

Practice Questions with Solutions

• Q: Solve 30x < 200 when (i) x is a natural number (ii) x is an integer. A: Step 1: Isolate x. 30x < 200 x < 200 / 30 x < 20 / 3 x < 6.66... Step 2: Apply conditions for x. (i) For x as a natural number: {1, 2, 3, 4, 5, 6} (ii) For x as an integer: {..., 0, 1, 2, 3, 4, 5, 6} Final answer: (i) {1, 2, 3, 4, 5, 6}, (ii) {..., 0, 1, 2, 3, 4, 5, 6}
• Q: Solve 4x + 3 < 5x + 7 for real x. A: Step 1: Move variable terms to one side. 4x - 5x + 3 < 7 -x + 3 < 7 Step 2: Move constant terms to the other side. -x < 7 - 3 -x < 4 Step 3: Isolate x (multiply by -1 and reverse sign). x > -4 Final answer: (-4, ∞)
• Q: Solve (x - 3) / 5 ≤ (x - 2) / 3 for real x. A: Step 1: Clear fractions by multiplying by LCM (15). 15 [(x - 3) / 5] ≤ 15 [(x - 2) / 3] 3(x - 3) ≤ 5(x - 2) 3x - 9 ≤ 5x - 10 Step 2: Collect variable terms. 3x - 5x - 9 ≤ -10 -2x - 9 ≤ -10 Step 3: Collect constant terms. -2x ≤ -10 + 9 -2x ≤ -1 Step 4: Isolate x (divide by -2 and reverse sign). x ≥ (-1) / (-2) x ≥ 1/2 Final answer: [1/2, ∞)
• Q: Find the solution of 7x + 3 ≤ 5x + 9 and show the graph of the solutions on a number line, when x is a real number. A: Step 1: Simplify the inequality. 7x + 3 ≤ 5x + 9 7x - 5x ≤ 9 - 3 2x ≤ 6 Step 2: Isolate x. x ≤ 6 / 2 x ≤ 3 Step 3: Represent on a number line. Draw a number line. Place a closed circle at '3' and shade the line to the left of '3'. Final answer: (-∞, 3]. Graph: Closed circle at 3, line shaded leftwards.

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