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Linear Inequalities

Action Plan Notes Questions Mock-Test

Study Action Plan (Sorted by Difficulty)

Feasible Region for Two-Variable Inequalities

Accuracy: 46%

Total Attempted13

Total Correct6

Total Incorrect7

Difficulty Level:

High

Applications of Inequalities

Accuracy: 50%

Total Attempted2

Total Correct1

Total Incorrect1

Difficulty Level:

Medium

Linear Inequalities in Two Variables

Accuracy: 70%

Learning Objectives

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Learning Objectives

Understand and solve linear inequalities in one variable using algebraic manipulation and represent solutions graphically on a number line.
Graphically solve linear inequalities in two variables using half-planes, boundary lines, and test points.
Comprehend the properties of inequalities, including the effects of addition, subtraction, and multiplication/division by positive or negative numbers on the inequality sign.
Graphically represent solutions of inequalities on a number line, distinguishing between strict and slack inequalities using open and closed circles.
Solve systems of inequalities and represent their solutions graphically, focusing on the intersection of solution sets.
Apply inequalities to solve real-world problems, such as determining minimum or maximum values under given constraints.
Solve double inequalities and express solutions in interval notation, such as solving $-8 \leq 5x - 3 < 7$ .
Identify the feasible region for two-variable inequalities, including boundary line inclusion/exclusion, for inequalities like $ax + by \leq c$ or $ax + by \geq c$ .

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Revision Notes & Summary

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Chapter Notes

Linear Inequalities in One Variable

Definition: An inequality involving a single variable, expressed in the form $ax + b < 0$ , $ax + b > 0$ , $ax + b \leq 0$ , or $ax + b \geq 0$ .
Example: Solve $30x < 200$ $30 x < 200$ .
- Solution:
  - Divide both sides by 30: $x < \frac{200}{30}$
  - Simplify: $x < \frac{20}{3}$
  - If $x$ is a natural number, solution set: {1, 2, 3, 4, 5, 6}
  - If $x$ is an integer, solution set: {-3, -2, -1, 0, 1, 2, 3, 4, 5, 6}

Linear Inequalities in Two Variables

Definition: An inequality involving two variables, expressed in the form $ax + by < c$ , $ax + by > c$ , $ax + by \leq c$ , or $ax + by \geq c$ .
Graphical Solution:
- Represented using half-planes and boundary lines.
- Use test points to determine which half-plane satisfies the inequality.

Properties of Inequalities

Addition/Subtraction: Equal numbers can be added to or subtracted from both sides without changing the inequality.
Multiplication/Division:
- By a positive number: Inequality sign remains the same.
- By a negative number: Inequality sign is reversed.

Graphical Representation of Inequalities

On a Number Line:
- Strict Inequality ( $<$ or $>$ ): Use open circles.
- Slack Inequality ( $\leq$ or $\geq$ ): Use closed circles.

System of Inequalities

Definition: A set of two or more inequalities with the same variables.
Solution: The intersection of the solution sets of the individual inequalities.
Graphical Representation: Use overlapping regions on a graph to find the common solution area.

Applications of Inequalities

Real-World Problems: Use inequalities to determine minimum or maximum values under given constraints.
Example: A student needs a minimum of 70 marks in the annual exam to achieve an average of at least 60 marks.

Double Inequalities and Interval Solutions

Example: Solve $-8 \leq 5x - 3 < 7$ $- 8 \leq 5 x - 3 < 7$ .
- Solution:
  - Split into two inequalities: $-8 \leq 5x - 3$ and $5x - 3 < 7$ .
  - Solve each separately and combine the solutions.
  - Solution set: $-1 \leq x < 2$ .

Feasible Region for Two-Variable Inequalities

Definition: The region that satisfies all given inequalities in a system.
Boundary Line Inclusion/Exclusion:
- Solid Line: Includes the boundary (inequality with $\leq$ or $\geq$ ).
- Dashed Line: Excludes the boundary (inequality with $<$ or $>$ ).

Examples

Example 1: Solve $4x + 3 < 6x + 7$ .
- Solution:
  - Rearrange: $-2x < 4$
  - Divide by -2: $x > -2$
  - Solution set: $(-2, \infty)$
Example 2: Solve $3x - 2 \geq x + 1$ .
- Solution:
  - Rearrange: $2x \geq 3$
  - Divide by 2: $x \geq \frac{3}{2}$
  - Solution set: $[\frac{3}{2}, \infty)$

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Practice Test – MCQs, True/False

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Multiple Choice Questions

₹3500

₹3800

₹4000

₹4200

Correct Answer: C

Solution:

Let

x

be the number of units of product A and

y

be the number of units of product B. The constraints are

x + y \leq 100

x \geq 10

, and

y \geq 10

. The profit function to maximize is

40x + 30y

. Testing the boundary condition

x + y = 100

, we find

40x + 30(100-x) = 4000 - 10x

. Maximizing this function subject to

x \geq 10

gives

x = 70

y = 30

, yielding a profit of

40(70) + 30(30) = 4000

Chapter Concept:

Applications of Inequalities

ax + b = 0

ax + by < c

ax^2 + bx + c = 0

ax + b > 0

Correct Answer: B

Solution:

A linear inequality in two variables is represented by expressions like

ax + by < c

, where

a

and

b

are not both zero.

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Linear Inequalities

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True or False

Correct Answer: True

Solution:

Multiplying or dividing both sides of an inequality by a negative number reverses the inequality sign, as shown by the example where

3 > 2

becomes

-3 < -2

Chapter Concept :

Graphical Representation of Inequalities

Correct Answer: True

Solution:

To solve the inequality

-8 \leq 5x - 3 < 7

, split it into two inequalities:

-8 \leq 5x - 3

and

5x - 3 < 7

. Solving the first gives

x \geq -1

and the second gives

x < 2

. Thus, the solution in interval notation is

[-1, 2)

Chapter Concept :

Double Inequalities and Interval Solutions

Correct Answer: False

Solution:

The inequality

30x < 200

represents a scenario where

x

must be a non-negative integer, as it denotes the number of packets of rice, which cannot be negative.

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Linear Inequalities

Study Action Plan (Sorted by Difficulty)

Feasible Region for Two-Variable Inequalities

High

Applications of Inequalities

Medium

Linear Inequalities in Two Variables

Learning Objectives

Learning Objectives

Practice, Analyze & Improve 🚀

Revision Notes & Summary

Chapter Notes

Linear Inequalities in One Variable

Linear Inequalities in Two Variables

Properties of Inequalities

Graphical Representation of Inequalities

System of Inequalities

Applications of Inequalities

Double Inequalities and Interval Solutions

Feasible Region for Two-Variable Inequalities

Examples

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Practice Test – MCQs, True/False

Multiple Choice Questions

Correct Answer: C

Q2.Which of the following represents a linear inequality in two variables?

Correct Answer: B

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True or False

Q1.When solving inequalities, multiplying or dividing both sides by a negative number reverses the inequality sign.

Correct Answer: True

Q2.Solving the compound inequality −8≤5x−3<7-8 \leq 5x - 3 < 7−8≤5x−3<7 and expressing the solution in interval notation results in the interval [−1,2)[-1, 2)[−1,2).

Correct Answer: True

Q3.The inequality 30x<20030x < 20030x<200 can have a solution set that includes negative integers.

Correct Answer: False

Medium

System of Inequalities

Medium

Properties of Inequalities

Low

Graphical Representation of Inequalities

Low

Linear Inequalities in One Variable

Low

Double Inequalities and Interval Solutions

Low

Q3.Solve the inequality 5x−3<3x+15x - 3 < 3x + 15x−3<3x+1 for xxx when xxx is a real number.

Correct Answer: A

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Q4.A farmer has a rectangular field with a perimeter of 200 meters. He wants to maximize the area of the field. What should be the dimensions of the field?

Correct Answer: A

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Q5.A student needs to maintain an average of at least 60 marks over three tests. If he scored 70 and 75 in the first two tests, what is the minimum score he must achieve in the third test?

Correct Answer: B

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Q6.In an experiment, a solution of hydrochloric acid is to be kept between 30° and 35° Celsius. What is the range of temperature in Fahrenheit?

Correct Answer: A

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Q7.Which of the following inequalities represents a feasible region that includes the origin?

Correct Answer: B

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Q8.If the inequality 4x+3<5x+74x + 3 < 5x + 74x+3<5x+7 is solved for xxx, which of the following represents the correct solution set?

Correct Answer: A

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Q9.If Ravi wants to buy packets of rice with a total cost less than ₹200, and each packet costs ₹30, what is the maximum number of packets he can buy?

Correct Answer: A

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Q10.Given the inequality 40x+20y≤12040x + 20y \leq 12040x+20y≤120, which of the following points lies within the feasible region?

Correct Answer: A