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Linear inequalities are types of algebraic expressions that are represented by linear functions. It involves the comparison of a polynomial of degree 1 to another algebraic expression of a degree less than or equal to 1.
- Inequalities can be numerical, algebraic, or a combination of both.
- Linear inequalities compare any two values by using inequality symbols like '<','>', '≤' or '≥'".
- They are considered similar to linear equations.
- Inequalities using the symbol'>' or '<' are called strict inequalities.
- Slack inequalities involve the use of symbols like ‘≤' or '≥'.
- The different solutions of linear inequalities are called solution sets.
- The statement involving the use of one variable is called a linear equation in one variable.
- Two-dimensional inequalities are represented in the form:
ax + by < c
- where a and b are real number and x is the variable.
Read More: Pair of Linear equation in two variables formula
Key Terms: Linear Inequalities, Linear Equation, Inequality, Polynomial Inequality, Rational Inequality, Absolute Value, Slack Inequalities, Strict Inequalities, Graph, Coordinate Plane
What is Linear Inequality in Maths?
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In mathematics, linear inequality denotes a mathematical expression in which neither side is equal. An inequality occurs when a relationship makes a non-equal comparison between two expressions or numbers.
- Any of the inequality symbols replace the equal sign "=" in the expression.
- Other symbols name includes greater than symbol (>), less than symbol (<) and less than or equal to symbol (≤).
- There are three types of inequalities: polynomial inequalities, rational inequalities, and absolute value inequalities.
- Polynomial is one quantity that is used in the linear inequalities.
- Inequalities can be created by use of one, two or more variable.
- Linear Inequalities can be used to determine the age of a particular person.
Read More: Linear Inequalities Important Questions
Some most commonly used linear inequality symbols:
| Symbol Name | Symbol | Example |
|---|---|---|
| Not equal | ≠ | x ≠ 3 |
| Less than | (<) | x + 7 < √2 |
| Greater than | (>) | 1 + 10x > 2 + 16x |
| Less than or equal to | (≤) | y ≤ 4 |
| Greater than or equal to | (≥) | -3 - √3x ≥ 10 |
Example of Linear InequalityExample 1: x > 3 is representation of linear inequality in one variable. Example 2: x + 7 < √2 is representation of linear inequality in two variables. Example 3: Suppose you want to buy a sandwich for yourself and you only have Rs. 5. So you will say to the that you would not pay more than Rs.5. So it can be represented by linear inequality: x ≤ 5 |
Read More: Pair of Linear equation in two variables Important Questions
How to Solve Linear Inequalities?
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To solve all types of linear inequalities, follow the steps below:
- Create an equation for the inequality first.
- Solve the given equation for one or more values.
- Represent all of the values obtained from the number line.
- On the number line, use open circles to represent the excluded values.
- Determine the interval.
- Choose a random value from the interval.
- Enter it into the inequality equation to see if the values satisfy the inequality equation.
- The solutions of the given inequality equation are intervals that satisfy the inequality equation.
Read More:
| Chapter Related Concept | ||
|---|---|---|
| Solving Inequalities | Geometric Progression | Permutations and Combinations |
| Trigonometric Functions | Linear Regression | Harmonic Progression |
Linear Inequalities Graphing
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The graph of an ordinary linear function after plotting for inequalities is observed here. However, the linear function graph is a line, whereas the graph of linear inequalities is the area of the coordinate plane that satisfies the inequality.
- A borderline divides the coordinate plane into two parts in the linear inequality graph.
- This is the line associated with the function.
- All solutions to inequality form one part of the borderline.
- The borderline is dashed for '>' and " inequalities and solid for " and ".
Read More: NCERT Solutions For Class 11 Maths Chapter 6: Linear Inequalities
Example for Linear Inequalities GraphingExample: To create an inequality graph for instance, y>x+2., we must first complete three steps:
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Read More: Addition of Two Real Functions
Rules of Linear Inequalities
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Addition, subtraction, multiplication, and division are the four operations performed on linear inequalities. Linear inequalities that have the same solution are referred to as equivalent inequality.
- Both equality and inequality have rules.
- All of the following rules apply to inequalities involving less than or equal to (), and greater than or equal to ().
Addition Rule of Linear Inequalities
In the addition rule of linear inequalities, same number is added to the given inequality that will produce the required linear inequality. It can be represented as:
a > y, then a + b > y + b
where b is the number added from the inequality.
Read More: Solving Inequalities
Subtraction Rule of Linear Inequalities
In the subtraction rule of linear inequalities, same number is subtracted to the given inequality that will produce the required linear inequality. It can be represented as:
a > y, then a - b > y - b
where b is the number subtracted from the inequality.
Multiplication Rule of Linear Inequalities
In the multiplication rule of linear inequalities, same number is multiplied to the given inequality that will produce the required linear inequality. It can be represented as:
a > y, then a x b > y x b
where b is the number multiplied from the inequality.
Division Rule of Linear Inequalities
In the division rule of linear inequalities, same number is divided to the given inequality that will produce the required linear inequality. It can be represented as:
a > y, then a / b > y / b
where b is the number divided from the inequality.
Solved Example of Rule of Linear InequalitiesExample 1: Solve 3x+8>2, when (i) x is an integer (ii) x is a real number Solution: Given Linear inequality: 3x+8>2 The given equation can also be written as 3x+8 -8 > 2 -8 …(1) Now, simplify the expression (1) ⇒ 3x > -6 Now, both sides, divide it by 3 ⇒ 3x/3 > -6/3 ⇒ x > -2 (i) x is an integer Hence, the numbers greater than -2 are -1,0,1,2,…etc Thus when x is an integer, the solutions of the given inequality are -1,0,1,2,… Hence, the solution set for the given linear inequality is {-1,0,1,2,…} (ii) x is a real number If x is a real number, the solutions of the given inequality are all the real numbers, which are greater than 2. Therefore, in the case of x being a real number, the solution set is (-2, ∞) Example 2: Ravi scored 70 and 75 marks in the first two-class test. Calculate the minimum marks he should get in the third test to have an average of at least 60 marks. Solution: Suppose that x is the mark obtained by Ravi in the third unit test. It is given that the student should have an average of at least 60 marks. From the given information, we can write the linear inequality as (70 + 75 + x)/3 ≥ 60 Now, simplify the expression: ⇒ (145 + x) ≥ 180 ⇒ x ≥ 180 -145 ⇒ x ≥ 35 Hence, Ravi should obtain a minimum of 35 marks to have an average of at least 60 marks. |
Read More: Horizontal and Vertical Lines
Things To Remember
- In linear inequalities, a relationship exists between LHS and RHS, such as less than or greater than.
- "Less than" and "greater than" are strict inequalities.
- A linear inequality is so named as the variable's highest power (exponents) is 1.
- A hollow dot represents the value obtained for x for each inequality.
- It demonstrates that the obtained value is ruled out.
- A solid dot represents the value obtained for x for any linear inequality that is not strict.
- It demonstrates that the obtained value is included.
- The method can be used to solve queries related to money and percentages.
Read More:
| Class 11 Mathematics Related Concept | ||
|---|---|---|
| Collinear Points | Straight Lines Formula | Cartesian Coordinate System |
| Number Theory | Sample Size | Binomial Expansion |
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Sample Questions
Ques.Explain Linear Inequalities in Algebra with examples? (2 marks)
Ans. Linear inequalities are expressions that compare two linear expressions using inequality symbols. These expressions could be numerical, algebraic, or a mix of the two.
- x - 5 > 3x - 10 is an example of a linear inequality.
- The greater than symbol is used in this inequality, the LHS is strictly greater than the RHS.
- After solving, the inequality is as follows: 2x<5 => x<(5/2).
Ques. What are the real-world applications of Linear Inequalities? (3 marks)
Ans. In many real-life problems, inequalities are used more frequently than equalities to determine the best solution. This solution can be as simple as determining how many units of a product should be manufactured to maximize profit. It can be as complex as determining the best combination of drugs to administer to a patient.
- Linear inequalities are used to solve speed and distance problems.
- It can be used to solve money related problems.
Ques. What Symbols are Used in Linear Inequalities? (2 marks)
Ans. The symbols used in linear inequalities are
- Not Equal(≠)
- Less than (<)
- Greater than (>)
- Less than or equal to (≤)
- Greater than or equal to (≥)
Ques. What are the two things that linear inequalities and equations have in common? (2 marks)
Ans. The similarities between linear inequalities and equations are:
- Both of the mathematical statements connect two expressions.
- They can be resolved in the same manner.
Ques. How to Solve Linear Inequalities in one and two variables? (3 marks)
Ans. One needs at least two inequalities to solve a system of two-variable linear inequalities. To find the solution, we will plot the given inequalities on a graph and look for a common shaded region.
- The inequality, we plot the given inequalities on the graph similarly to linear equations, but with dotted lines.
- To find the solution, we look for the common shaded region.
- If no common shaded region exists, the solution does not exist.
- The shaded area can be either bounded or unbounded.
Ques. What distinguishes quadratic inequalities from linear inequalities? (1 mark)
Ans. Linear inequalities are composed of algebraic expressions of degree 1, whereas quadratic inequalities are composed of algebraic expressions of degree 2.
Ques. Solve the inequality 4 ( x + 2 ) − 1 > 5 − 7 ( 4 − x )? (4 marks)
Ans. Given,
4 ( x + 2 ) − 1 > 5 − 7 ( 4 − x )
Expanding the brackets and multiplying by each term we get;
4 x + 8 − 1 > 5 − 28 + 7 x
4 x + 7 > − 23 + 7 x
Subtract 7 on both the sides
4x + 7 – 7 > -23 + 7x – 7
4x > -30 + 7x
Subtracting 7x from both the sides
4x – 7x > -30 + 7x – 7x
− 3 x > − 30
Multiplying both sides by -1, the inequality gets reversed;
-3x (-1) < -30 x (-1)
3x < 30
Dividing both sides by 3, we get;
3x/3 < 30/3
x < 10
Hence, x lies between -∞ and less than 10.
Ques: Solve the following inequality 8 ≤ 3 − 5 x < 12? (4 Marks)
Ans: Given,
8 ≤ 3 − 5 x < 12
We can distribute the intervals as
8 ≤ 3 – 5x and 3 – 5x < 12
Now taking one at a time.
8 ≤ 3 – 5x
Subtracting 3 on both the side
8 – 3 ≤ 3 – 5x – 3
5 ≤ -5x
Multiplying by (-1) on both sides reverses the inequality sign.
5 (-1) ≥ -5x (-1)
-5 ≥ 5x
Dividing both sides by 5, we get;
-5/5 ≥ 5x/5
-1 ≥ x … (i)
Now, taking the second interval, we have;
3 – 5x < 12
Subtracting 3 on both sides;
3 – 5x – 3 < 12 – 3
-5x < 9
Multiplying by (-1) on both sides reverses the inequality sign.
(-1) (-5x) > 9 (-1)
5x > -9
Dividing by 5 on both sides;
5x/5 > -9/5
x > -9/5 … (ii)
On combining both the intervals (i) and (ii), we can write;
− 1 ≥ x > − 9/5
or
-9/5 < x ≤ -1
Hence, x lies between the interval (-9/5,-1).
Ques. Solve 4x+8>16, when (A) x is an integer
(B) x is a real number? (4 marks)
Ans. Given Linear inequality: 4x+8>16
The given equation can also be written as
4x+8 – 8 > 16 -8 …(1)
Now, simplify the expression (1)
⇒ 4x > 8
Now, both sides, divide it by 4
⇒ 4x/4 > 8/4
⇒ x > 4
(A) x is an integer
Hence, the numbers greater than 4 are 5,6,7,…etc
Thus when x is an integer, the solutions of the given inequality are 5,6,7,…etc
Hence, the solution set for the given linear inequality is {5,6,7...}
(B) x is a real number
If x is a real number, the solutions of the given inequality are all the real numbers, which
are greater than 4.
Therefore, in the case of x being a real number, the solution set is (4, ∞)
Ques. Ravi scored 80 and 85 marks in the first two-class test. Calculate the minimum marks he should get in the third test to have an average of at least 70 marks? (3 marks)
Ans. Suppose that x is the mark obtained by Ravi in the third unit test.
It is given that the student should have an average of at least 60 marks.
From the given information, we can write the linear inequality as
(80 + 85 + x)/3 ≥ 70
Now, simplify the expression:
⇒ (165 + x) ≥ 210
⇒ x ≥ 210 -165
⇒ x ≥ 45
Hence, Ravi should obtain a minimum of 45 marks to have an average of at least 70 marks.
Ques. Solve the inequality 6 ( x + 2 ) − 1 > 4 − 7 ( 3 − x )? (4 marks)
Ans. Given,
6 ( x + 2 ) − 1 > 4 − 7 ( 3 − x )
Expanding the brackets and multiplying by each term we get;
6 x + 12 − 1 > 4 − 21 + 7 x
6 x + 11 > − 17 + 7 x
Subtract 11 on both the sides
4x + 11 – 11 > -23 + 7x – 11
4x > -34 + 7x
Subtracting 7x from both the sides
4x – 7x > -34 + 7x – 7x
− 4x > − 34
Multiplying both sides by -1, the inequality gets reversed;
-4x (-1) < 34 x (-1)
4x < 34
Dividing both sides by 4, we get;
4x/4 < 34/4
x < 8.5
Hence, x lies between -∞ and less than 8.5.
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