ICSE Class 10 Maths Chapter 05 Linear Inequations

Unit 2 Algebra

Chapter 5 Linear Inequations

Points To Remember

1. Linear Inequation: A statement of inequality between two expressions involving a single variable x with highest power one, is called a linear inequation. The general forms of linear inequations are:

(i) \(ax + b > c\)

(ii) \(ax + b < c\)

(iii) \(ax + b \geq c\)

(iv) \(ax + b \leq 0\)

where a, b, c are real numbers and \(a \neq 0\).

2. Replacement Set or Domain of the Variable: The set from which the values of the variable x are replaced in an inequation, is called the replacement set or the domain of the variable. The replacement set is always given to us.

3. Solution Set: The set of all those values of x from the replacement set which satisfy the given inequation, is called the solution set of the inequation. Solution set is always a subset of the replacement set.

4. Properties of Inequations:

(i) Adding or subtracting the same number or expression to each side of an inequation does not change the inequality.

(ii) Multiplying or dividing each side of an inequation by the same positive number does not change the inequality.

(iii) Multiplying or dividing each side of an inequation by the same negative number reverses the inequality.

(iv) \(a < b \Leftrightarrow b > a\)

(v) \(a > b \Leftrightarrow b < a\)

5. Some Special Sets of Numbers shown on Number Line:

We should like to have a glimpse on how we represent sets of numbers on a number line.

Teacher's Note

Understanding inequations helps in real-world scenarios like determining safe speed limits or budget constraints in daily decision-making.

Exercise 5

Solve each of the inequations given below and represent its solution set on a number line:

Q.1. \(2x - 7 < 4, x \in \{1, 2, 3, 4, 5, 6, 7\}\)

Sol. \(2x - 7 < 4\)

\(\Rightarrow 2x < 4 + 7\)

\(\Rightarrow 2x < 11 \Rightarrow x < \frac{11}{2}\)

\(\therefore\) Solution set is \(\{1, 2, 3, 4, 5\}\) Ans.

Q.2. \(2x - 3 > 3, x \in \{1, 2, 3, 4, 5, 6\}\)

Sol. \(2x - 3 > 3\)

\(\Rightarrow 2x > 3 + 3\)

\(\Rightarrow 2x > 6 \Rightarrow x > 3\)

\(\therefore\) Solution set is \(\{4, 5, 6\}\) Ans.

Q.3. \(9 \leq 1 - 2x, x \in \{-3, -4, -5, -6\}\)

Sol. \(9 \leq 1 - 2x\)

\(\Rightarrow 2x \leq 1 - 9\)

\(\Rightarrow 2x \leq -8 \Rightarrow x \leq \frac{8}{2}\)

\(\Rightarrow x \leq -4\)

\(\therefore\) Solution set is \(\{-4, -5, -6\}\) Ans.

Q.4. \(\frac{3x - 5}{6} > \frac{1}{2}, x \in \{0, 1, 2, 3, 4, 5, 6\}\)

Sol. \(\frac{3x - 5}{6} > \frac{1}{2}\)

\(\Rightarrow 3x - 5 > \frac{1}{2} \times 6\)

\(\Rightarrow 3x - 5 > 3\)

\(\Rightarrow 3x > 3 + 5\)

\(\Rightarrow 3x > 8 \Rightarrow x > \frac{8}{3}\)

\(\Rightarrow x > 2\frac{2}{3}\)

\(\therefore\) Solution set is \(\{3, 4, 5, 6\}\) Ans.

Q.5. \(7x - 4(3 - x) \geq 3(2x - 5), x \in \{-3, -2, -1, 0, 1, 2, 3\}\)

Sol. \(7x - 4(3 - x) \geq 3(2x - 5)\)

\(\Rightarrow 7x - 12 + 4x \geq 6x - 15\)

\(\Rightarrow 7x + 4x - 6x \geq -15 + 12\)

\(\Rightarrow 5x \geq -3 \Rightarrow x \geq -\frac{3}{5}\)

\(\therefore\) Solution set is \(\{0, 1, 2, 3\}\) Ans.

Q.6. \(11 - 3x > 2 + x, x \in \{1, 2, 3, 4, 5, 6\}\)

Sol. \(11 - 3x > 2 + x\)

\(\Rightarrow -3x - x > 2 - 11\)

\(\Rightarrow -4x > -9\)

\(\Rightarrow 4x < 9 \Rightarrow x < \frac{9}{4}\)

\(\Rightarrow x < 2\frac{1}{4}\)

\(\therefore\) Solution set is \(\{1, 2\}\) Ans.

Teacher's Note

Solving inequations step-by-step is like following a recipe - each operation must be applied correctly to reach the right answer, just as missing a step in cooking ruins the dish.

Q.7. \(4 - 3x \geq 3x - 14, x \in \mathbb{N}\)

Sol. \(4 - 3x \geq 3x - 14\)

\(\Rightarrow -3x - 3x \geq -14 - 4\)

\(\Rightarrow -6x \geq -18 \Rightarrow 6x \leq 18\)

\(\Rightarrow x \leq \frac{18}{6} \Rightarrow x \leq 3\)

\(\therefore\) Solution set is \(\{1, 2, 3\}\) where \(x \in \mathbb{N}\) Ans.

Q.8. \(6 - 5x > 3 - 4x, x \in \mathbb{W}\)

Sol. \(6 - 5x > 3 - 4x\)

\(\Rightarrow -5x + 4x > 3 - 6\)

\(\Rightarrow -x > -3 \Rightarrow x < 3\)

\(\therefore\) Solution set is \(\{0, 1, 2\}\) where \(x \in \mathbb{W}\) Ans.

Q.9. \(30 - 2(3x - 4) < 24, x \in \mathbb{W}\)

Sol. \(30 - 2(3x - 4) < 24\)

\(\Rightarrow 30 - 6x + 8 < 24\)

\(\Rightarrow -6x < 24 - 30 - 8\)

\(\Rightarrow -6x < 24 - 38 \Rightarrow -6x < -14\)

\(\Rightarrow 6x > 14 \Rightarrow x > \frac{14}{6}\)

\(\Rightarrow x > 2\frac{1}{3}\)

\(\therefore\) Solution set is \(\{3, 4, 5, 6, \ldots\}\), where \(x \in \mathbb{W}\) Ans.

Q.10. \(\frac{3}{5}x - \frac{2x - 1}{3} > 1, x \in \mathbb{I}\)

Sol. \(\frac{3}{5}x - \frac{2x - 1}{3} > 1\)

Multiplying by 15, the LCM of 5 and 3

\(15 \times \frac{3}{5}x - 15 \times \frac{2x - 1}{3} > 1 \times 15\)

\(\Rightarrow 9x - 5(2x - 1) > 15\)

Teacher's Note

Working with fractions in inequations requires finding the LCM, similar to how we need a common denominator in cooking measurements when combining ingredients.

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