Linear Inequalities

Linear inequalities are algebraic expressions where the power of the unknown variable is no more than one, and the variable is connected with an inequality sign (>, <, ≤, or  ≥).   

7x – 12 > 16 and 5x + 11 < 2 are examples of linear inequalities

Rules to Solve

For Adding or Subtracting 

Adding or subtracting a number on or from both sides of the inequality does not change its direction. 

If a < b, then a + c < b + c

If a > b, then a + c > b + c

Similarly,

If a < b, then a – c < b – c

If a > b, then a – c > b – c

For Multiplying or Dividing

Like adding and subtracting, multiplying, or dividing an inequality by the same positive number also does not change the direction of the inequality. 

If a < b and c > 0, then ac < bc

If a > b and c > 0, then ac > bc

If a < b and c < 0, then ac > bc

If a > b and c < 0, then ac < bc

Similarly,

If a < b and c > 0, then ${\dfrac{a}{c} <\dfrac{b}{c}}$

If a > b and c > 0, then ${\dfrac{a}{c} >\dfrac{b}{c}}$

If a < b and c < 0, then ${\dfrac{a}{c} >\dfrac{b}{c}}$

If a > b and c < 0, then ${\dfrac{a}{c} <\dfrac{b}{c}}$

Solving and Graphing

When solving a linear inequality, the solution is typically represented as an ordered pair (x, y) that satisfies the inequality, which is then graphed on a number line.

One-Step

Using the above rules, we solve the inequality x + 3 > 10

Step 1: Using the Subtraction Property

x + 3 – 3 > 10 -3

⇒ x > 7

Step 3: Graphing the Solution

Two-Step

For One Variable

Let us try solving the inequality with one variable as 4x + 3 < 23

Step 1: Using the Subtraction Property

4x + 3 – 3 < 23 – 3

⇒ 4x < 20

Step 2: Using the Division Property

${\dfrac{4x}{4} <\dfrac{20}{4}}$

⇒ x < 5

Thus, the solution is x < 5

Step 3: Graphing the Solution

For Two Variables:

Now, we solve the inequality 7y – 5x ≤ 6y – 3x + 3

Step 1: Solve for ‘x’ Using the Addition Property

7y – 5x + 5x ≤ 6y – 3x + 5x + 3

⇒ 7y ≤ 6y + 2x + 3

Step 2: Solve for ‘y’ Using the Subtraction Property

7y – 6y ≤ 6y + 2x + 3 – 6y

⇒ y ≤ 2x + 3

Step 3: Graphing the Solution

To plot a graph of the inequality y ≤ 2x + 3, we consider ‘≤’ as ‘=’ sign. Now, plotting the graph y = 2x + 3, we get

Since the inequality is ‘≥,’ the graph of the equation is formed as a solid line. 

Now, considering a point (x, y) as (-2, 0), we get 

2x + 3 = 2(-2) + 3 = -4 + 3 = -1, which is not greater than 0

Thus, y ≰ 2x + 3 at (-2, 0)

Now, shading the region that does not contain (-2, 0) shows the following graph.

Solved Examples

Special Cases

Sometimes, plotting the solution of linear inequalities can form a horizontal line parallel to the x-axis or a vertical line parallel to the y-axis.

Case 1: When we plot the graph of the linear inequality y > 1 (‘y’ is greater than 1, excluding 1), the graph includes the entire region above the line y = 1, without the line y = 1 (shown in dotted form). 

Case 2: When we plot the graph of the linear inequality x < 4 (‘x’ is less than 4, excluding 4), the graph includes the entire region left of line x = 4, without the line x = 4 (shown in dotted form).

System of Linear Inequalities

A system of linear inequalities consists of two or more linear inequalities with the same variables. Its solution includes all ordered pairs that simultaneously satisfy each inequality in the system.

Graphing

Graphically, the solution set is depicted as the intersection of the regions represented by each individual inequality on the coordinate plane. 

Let us consider the system of inequalities:

5y – 3x < 15 and 5x + 3y > 15

First, we graph each inequality separately, as shown.

Now, let us pick a point that is not on the two given lines and verify whether this point satisfies the inequalities. 

Considering the point (x, y) as (2, 1), we get 

5y – 3x = 5(1) – 3(2) = 5 – 6 = -1, which is less than 15

5x + 3y = 5(2) + 3(1) = 10 + 3 = 13, which is not greater than 15

Thus, the individual shaded regions and their overlapped region are shaded:

Here, the graphs are formed as dotted lines for the strict inequalities (‘<’ and ‘>’).