How to Solve Inequality Problems

In mathematics, an inequality is a mathematical expression that has two sides that are not equal. In mathematics, inequality occurs when a connection makes a non-equal comparison between two expressions or two numbers. In this situation, the inequality symbols greater than symbol (>), less than symbol (<), greater than or equal to symbol (≥), less than or equal to symbol (≤), or not equal to symbol (≠) substitute the equal sign “=” in the expression. Polynomial inequality, rational inequality, and absolute value inequality are examples of inequalities in mathematics.

Strict inequalities are represented by the symbols < and >, while slack inequalities are represented by the symbols ≤ and ≥. A linear inequality looks precisely like a linear equation, but the symbol that connects two expressions is different.

Linear equality

The graphical approach of solving linear inequalities is a simple way to obtain solutions for linear equations. It’s easy to solve a one-variable linear equation by plotting the value on a number line. For two-variable scenarios, however, the graph must be plotted in an x-y plane. A linear function is used in linear inequality. An equation is a mathematical expression that contains the equal-to (=) symbol. The equality symbol indicates that the right-hand side of the phrase is the same as the left-hand side. Inequalities are created when two mathematical expressions contain symbols such as greater (>), less than(<), greater than or equal (≥), less than or equal (≤).

For instance,

Statement 1— Neeta is 15 years old.

The mathematical expression for the equality symbol is x = 15.

Statement 2 — If I assert that Neeta is at least 15 years old, then this can be written as x ≥ 15.

As a result, Statement 1 is an equation, while Statement 2 is an inequality.

Solving Linear Inequality

Let’s talk about how to solve inequalities graphically immediately. A straight line is the graph of a linear equation, and any point in the Cartesian plane with regard to it will lie on either side of it. Consider the expression ax + by for a two-variable linear equation. This statement can be used to represent the following inequalities.

1. ax + by < c

2. ax + by ≤ c

3. ax + by > c

4. ax + by ≥ c

Consider the line ax + by = c, a ≠ 0 and b ≠ 0 to solve these inequalities. This line’s graph may be seen below: 

The graph of the line splits the Cartesian plane into two parts. All points (x, y) on the line axe + by = c satisfy the equation, indicating that both sides of the equation are equal. A solution to the line is represented by the point (α,β) on the line ax + by = c. Thus, aα + bβ = c.

Consider the upper half of the plane and the N(α, ϒ). The graph clearly indicates that ϒ >β. Also, because the graph is represented as a straight line, it indicates that b>0; otherwise, the equation ax + by = c would just be a point.

⇒ bϒ > bβ

⇒ aα + bϒ > aα + bβ 

⇒ aα + bϒ > c

Any point Q(α, ϒ) above or in the upper half of the plane, or the region denoted by I produced by the line ax + by = c, would meet the inequality ax + by > c.

Linear Inequalities Graphing

• ax + by < 0 and ax + by > 0

The region designated II, which is below the line ax + by = c, contains all the points that satisfy the inequality ax + by < c. As a result, zone II is the solution region for the inequality of type ax + by < c. The line ax + by = c is dotted because it is outside of the solution zone.

All of the points that fulfil the inequality ax + by > c are found in the top half of the line ax + by = c, or the region indicated I. As a result, Zone I is the solution region for the inequality ax + by > c. The line ax + by = c is dotted because it is outside of the solution zone.

• ax + by ≤ 0 and ax + by ≥ 0

All points that meet the inequality ax + by ≤ c are found in the region labelled II, which is below and includes the line ax + by = c. As a result, Zone II is the solution region for the inequality of type ax + by ≤ c.

All points that meet the inequality ax + by ≥ c are found in the region below and including the line ax + by = c, or the region labelled I. As a result, Zone I is the solution region for the inequality of type ax + by ≥ c.

Conclusion

Therefore we can finally conclude that strict inequalities are represented by the symbols < and >, while slack inequalities are represented by the symbols ≤ and ≥. A linear inequality looks precisely like a linear equation, but the symbol that connects two expressions is different. The graphical approach of solving linear inequalities is a simple way to obtain solutions for linear equations. It’s easy to solve a one-variable linear equation by plotting the value on a number line. For two-variable scenarios, however, the graph must be plotted in an x-y plane. A linear function is used in linear inequality.