Linear Inequation

Important Questions

Class 10 Linear inequalities are a fundamental topic in mathematics, playing a pivotal role in solving real-world problems and understanding mathematical concepts. A solid grasp of linear inequalities is essential for success in ICSE Class 10 mathematics. In this collection of important questions, we will delve into the core concepts of linear inequalities, covering a wide range of problem types. These linear inequations class 10 ICSE questions are designed to help you practice and prepare effectively, ensuring that you are well-equipped to handle linear inequalities with confidence.

Solving Linear Inequalities:

a. Solve the inequality: 2x - 5 < 7.

b. Determine the solution set for 3(x + 4) ≥ 15.

c. Solve the inequality: 2(x - 3) < 4x + 5.

Graphical Representation:

a. Graph the inequality: 2x + 3y ≤ 6.

b. Illustrate the solution set for the inequality: x > -2.

Tables of Content

A linear inequality in linear inequations class 10 icse is an expression involving linear functions (polynomials of degree 1) and comparison operators (such as <, >, ≤, or ≥). Unlike linear equations, which equate two expressions, linear inequalities establish a relationship between two expressions, indicating that one is less than, greater than, less than, or equal to, or greater than or equal to the other. Linear inequalities can be visually represented on a coordinate plane. The solutions lie either above or below a boundary line, depending on the inequality's direction. Shading regions on the graph corresponding to the solution set provide a clear visualization of the valid solutions.

linear inequalities, often referred to as "linear inequations," are mathematical expressions or statements that involve linear functions and a comparison operator to establish a relationship between two mathematical expressions, indicating that one is less than, greater than, less than, or equal to, or greater than or equal to the other. These linear inequations are essential in modeling real-world situations with inequalities and constraints.

Here are the key components and concepts related to linear inequations class 10 ICSE:Linear Functions: Linear functions are algebraic expressions that consist of constants, variables raised to the first power, and the basic operations of addition, subtraction, multiplication, and division. For example, expressions like 2x + 3, 4y - 1, and 0.5z are linear functions.

Comparison Operators: Linear inequalities use comparison operators to define the relationship between two linear expressions. The commonly used operators are:

< (less than)

(greater than)

≤ (less than or equal to)

≥ (greater than or equal to)

Linear Inequality Expressions: A linear inequality expression is typically written in the form:

Linear Expression_1 <comparison operator> Linear Expression_2

For example, 2x + 3 > 7 is a linear inequality where 2x + 3 and 7 are the linear expressions, and > is the comparison operator.Solving Linear Inequalities:

Solving a linear inequality involves finding the values of the variable(s) that satisfy the inequality. The set of all valid solutions is called the "solution set." To solve linear inequalities, you may need to manipulate the inequality by performing operations like addition, subtraction, multiplication, or division on both sides, just like solving equations. However, keep in mind that when you multiply or divide by a negative number, the direction of the inequality sign may change.

(b) {– 2, – 1, 0, 1, 2, 3, 4}

(c) {– 3, – 2, – 1, 0, 1, 2, 3, 4, 5}

(d) {– 3, – 2, – 1, 0, 1, 2, 3, 4}

**Ans**. (d) {– 3, – 2, – 1, 0, 1, 2, 3, 4}

**Explanation:**We have

2x-5 ≤ 5x+4 < 29, x ∈ I

⇒ 2x-5 ≤ 5x+4; 5x + 4 < 29

⇒ 2x - 5x ≤ 4 + 5; 5x < 29 - 4

⇒ - 3x ≤ 9; 5x < 25

⇒ \(\frac{-3x}{-3}\) ≥ \(\frac{9}{-3}\); \(\frac{5x}{5}\) < \(\frac{25}{5}\)

⇒ x ≥ - 3; x < 5

∴ x = {– 3, – 2, – 1, 0, 1, 2, 3, 4}

(b) {x : x ∈ R, –2 < x ≤ 3}

(c) {x : x ∈ R, –2 ≤ x ≤ 3}

(d) {x : x ∈ R, –2 < x < 3}

**Ans**. (b) {x : x ∈ R, –2 < x ≤ 3}

**Explanation:**{x : x ∈ R, –2 < x ≤ 3}

The adjacent figure shows

x > - 2 and x ≤ 3

So, answer will be {x : x ∈ R, -2 < x ≤ 3}

**Explanation:**\(\frac{-x}{3}\)

-2x

8

\(\frac{8}{5}\)

\(\frac{8}{5}\)

Set = {x : x ∈ R, \(\frac{8}{5}\)

\(\frac{x}{3}\) ≤ \(\frac{x}{3}\) 1\(\frac{1}{3}\) <\(\frac{1}{6}\), x ∈ R

**Explanation:**we have \(\frac{x}{3}\) ≤ \(\frac{x}{3}\) 1\(\frac{1}{3}\) <\(\frac{1}{6}\), x ∈ R

Now,

Thus, \(\frac{8}{5}\) ≤ x < 3

Or 1.6 ≤ x < 3

∴ Solution set = [ x : 1.6 ≤ x < 3, ∈ R ].

\(\frac{3x+15}{5}\)

**Explanation:**Taking L.C.M. of 5, 2 = 10 and on multiplying throughout by 10, we get:

\(\frac{3x+15}{5}\) × 10 ≤ (2x + 3) × 10 ≤ \(\frac{3x+15}{2}\) × 10

2(3x+15) ≤ 10(2x+3) ≤ 5(3x+15), which on simplification gives:

6x + 30 ≤ 20x + 30 ≤ 15x + 75,

Solving the LHS part: 6x + 30 ≤ 20x + 30

⇒ 14x ≥ 0

⇒ x ≥ 0...........(i)

Again, solving the RHS part: 10(2x+3) ≤ 5(3x+15)

⇒ 20x + 30 ≤ 15x + 75

⇒ 20x - 15x ≤ 75 - 30

⇒ 5x ≤ 45

⇒ x ≤ \(\frac{45}{5}\)

⇒ x ≤ 9 ...........(ii)

In conclusion, linear inequations of linear inequations class 10 ICSE are a fundamental concept of Mathematics. They provide a powerful tool for modeling and solving real-world problems that involve inequalities, constraints, and decision-making. If you wish to improve your skills further and gain a better grasp of the concepts covered in this chapter, you can explore oswal.io, where you'll find a wide range of practice questions. These linear inequations class 10 ICSE questions are designed to assist you in your journey to achieve a more in-depth understanding of the material.

**Ans: **A linear inequality is a mathematical statement that involves linear functions and comparison operators to represent relationships between two expressions, indicating that one is less than, greater than, less than, or equal to, or greater than or equal to the other.

**Ans: **To solve a linear inequality, perform operations like addition, subtraction, multiplication, or division on both sides while considering the direction of the inequality sign. The result will yield the valid values for the variable (s).

**Ans: **The solution set of a linear inequality is the set of all values of the variable (s) that satisfy the inequality, making it true. It can be represented as a range of values.

**Ans:** To graphically represent a linear inequality, plot the boundary line (usually a straight line) corresponding to the equation formed by the inequality. Then, shade the region above or below the line to indicate the solution set.

**Ans:** Yes, a linear inequality can have no solution if there is no range of values for the variable (s) that satisfies the inequality. This occurs when the solution region on the graph is empty.