This chapter introduces you to linear inequalities, which are mathematical statements that compare two expressions using inequality signs like <, >, ≤, or ≥. Unlike equations, inequalities do not require the two sides to be equal but rather express a range of values that satisfy the given condition. You will learn how to solve these inequalities algebraically and graphically, both in one and two variables. This chapter is crucial as it lays the foundation for solving real-world problems in various fields such as science, economics, and statistics, where exact equality is not always possible or necessary.
What you'll learn
1Understand what linear inequalities are and how they differ from equations
2Solve linear inequalities in one variable algebraically
3Represent the solutions of linear inequalities on a number line
4Graph linear inequalities in two variables
5Solve systems of linear inequalities in two variables
6Apply the concepts of linear inequalities to real-world problems
Chapter at a glance
01Introduction to Linear Inequalities
02Inequalities
03Algebraic Solutions of Linear Inequalities in One Variable
04Graphical Solution of Linear Inequalities in One Variable
05Linear Inequalities in Two Variables
065.3 Algebraic Solutions of Linear Inequalities in One Variable and their Graphical Representation
07Solutions of System of Linear Inequalities in Two Variables
Detailed chapter notes
01
Introduction to Linear Inequalities
Linear inequalities are mathematical statements that compare two expressions using inequality signs such as less than (<), greater than (>), less than or equal to (≤), and greater than or equal to (≥). These inequalities are useful in situations where exact equality is not required or possible. For example, if the height of all students in a class is less than 160 cm, this can be represented as an inequality. This chapter focuses on linear inequalities in one and two variables, which are essential for solving problems in various fields like science, mathematics, and economics.
Inequalities compare two expressions using <, >, ≤, or ≥
They are useful in real-world problems where exact equality is not necessary
02
Inequalities
An inequality involves two real numbers or algebraic expressions connected by an inequality sign. For example, 3 < 5 is a numerical inequality, while x < 5 is a literal inequality. Double inequalities, such as 3 < x < 5, combine two inequalities. Linear inequalities in one variable (like ax + b < 0) and two variables (like ax + by < c) are particularly important. The chapter focuses on solving these linear inequalities, both strict (e.g., ax + b < 0) and slack (e.g., ax + b ≤ 0).
Numerical inequalities3 < 5, 7 > 5
Literal inequalitiesx < 5, y > 2, x ≥ 3, y ≤ 4
Double inequalities3 < x < 5, 2 < y < 4
Linear inequalities in one variableax + b < 0, ax + b > 0, ax + b ≤ 0, ax + b ≥ 0
Linear inequalities in two variablesax + by < c, ax + by > c, ax + by ≤ c, ax + by ≥ c
03
Algebraic Solutions of Linear Inequalities in One Variable
To solve a linear inequality in one variable, you need to find all values of the variable that make the inequality true. The rules for solving inequalities are similar to those for solving equations, with one key difference: when you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign. For example, solving 30x < 200 involves dividing both sides by 30, resulting in x < 20/3. The solution set includes all real numbers less than 20/3.
Rule 1Equal numbers may be added to or subtracted from both sides of an inequality without changing the inequality sign
Rule 2Both sides of an inequality can be multiplied or divided by the same positive number. If the number is negative, reverse the inequality sign
04
Graphical Solution of Linear Inequalities in One Variable
Graphical representation of solutions to linear inequalities in one variable involves plotting the solution set on a number line. For example, the solution to x < 3 is represented by a line extending to the left from 3, with an open circle at 3 to indicate that 3 is not included. Similarly, the solution to x ≥ 1 is represented by a line extending to the right from 1, with a closed circle at 1 to indicate that 1 is included. This visual representation helps in understanding the range of values that satisfy the inequality.
Use a number line to represent the solution set
An open circle indicates that the endpoint is not included (strict inequality)
A closed circle indicates that the endpoint is included (non-strict inequality)
05
Linear Inequalities in Two Variables
Linear inequalities in two variables, such as ax + by < c, can be represented graphically on a coordinate plane. The solution set includes all points (x, y) that satisfy the inequality. To graph the inequality, first draw the line corresponding to the equation ax + by = c. If the inequality is strict (e.g., < or >), use a dashed line to indicate that the points on the line are not included in the solution set. If the inequality is non-strict (e.g., ≤ or ≥), use a solid line. The solution set is the region of the plane that satisfies the inequality, which can be determined by testing a point not on the line.
Graph the line corresponding to the equation ax + by = c
Use a dashed line for strict inequalities (<, >) and a solid line for non-strict inequalities (≤, ≥)
Shade the region that satisfies the inequality based on a test point
06
Solutions of System of Linear Inequalities in Two Variables
A system of linear inequalities in two variables consists of two or more inequalities that must be satisfied simultaneously. The solution set of the system is the intersection of the solution sets of the individual inequalities. Graphically, this is the region where all the shaded areas overlap. For example, solving the system 3x – 7 < 5 + x and 11 – 5x ≤ 1 involves finding the values of x that satisfy both inequalities. The solution set is the range of x values that lie within the overlapping regions of the individual solution sets, which can be represented on a number line.
Find the intersection of the solution sets of individual inequalities
Graphically, the solution set is the overlapping shaded region
Represent the solution on a number line for clarity
Want the complete chapter resources?Topic notes, quizzes and flashcards for Linear Inequalities.
Which of the following is a linear inequality in one variable?
A2x + 3 = 7
B2x + 3 > 7
Cx² + 2x + 1 = 0
Dxy + 2y = 5
Show answer
Answer: (B) 2x + 3 > 7
A linear inequality contains a linear expression and an inequality sign (>, <, ≥, ≤). Option b is the only one with a linear expression and an inequality sign.
Question 02
Which of the following is a linear inequality in one variable?
A2x + 3 = 7
B2x + 3 > 7
Cx² + 2x + 1 = 0
Dxy + 2 < 5
Show answer
Answer: (B) 2x + 3 > 7
A linear inequality in one variable contains only one variable with power 1 and uses inequality symbols (>, <, ≥, ≤) instead of equality. Option b satisfies both conditions.
Question 03
Solve the inequality: 2x + 3 > 7
Ax > 2
Bx < 2
Cx ≥ 2
Dx ≤ 2
Show answer
Answer: (A) x > 2
Subtracting 3 from both sides gives 2x > 4, then dividing by 2 yields x > 2.
Question 04
The graphical solution of the inequality x > 5 on a number line is represented by which of the following?
AA closed circle at 5 with shading to the right
BAn open circle at 5 with shading to the right
CA closed circle at 5 with shading to the left
DAn open circle at 5 with shading to the left
Show answer
Answer: (B) An open circle at 5 with shading to the right
For strict inequality x > 5, we use an open circle (not including 5) and shade to the right of 5 on the number line.
Question 05
Which of the following is a linear inequality in two variables?
A2x + 3y > 5
Bx² + y = 4
C3x + 2y + z ≤ 10
Dxy + 2 < 8
Show answer
Answer: (A) 2x + 3y > 5
A linear inequality in two variables has the form ax + by + c > d (or <, ≤, ≥) where x and y appear with power 1. Option a satisfies this definition; others contain exponents, three variables, or products of variables.
Ready for more practice?Unlock the full quiz for this chapter.
What is the difference between a strict and a slack inequality?
A strict inequality uses < or > and does not include the boundary point, while a slack inequality uses ≤ or ≥ and includes the boundary point.
How do you represent the solution of x < 3 on a number line?
Draw an open circle at 3 and shade the line to the left to indicate all numbers less than 3.
What happens to the inequality sign when you multiply both sides by a negative number?
The inequality sign is reversed. For example, if you multiply both sides of x < 5 by -1, the inequality becomes -x > -5.
How do you graph the inequality y > 2x + 1 on a coordinate plane?
First, draw the line y = 2x + 1 as a dashed line. Then, shade the region above the line to indicate all points where y is greater than 2x + 1.
What is the solution set of a system of linear inequalities?
The solution set of a system of linear inequalities is the set of all points that satisfy all the inequalities simultaneously, which is the intersection of the individual solution sets.
Why are linear inequalities important in real-world problems?
Linear inequalities are important because they allow us to model situations where exact equality is not required or possible, such as in budgeting, resource allocation, and optimization problems.