Solving Inequality Worksheets

In mathematics, inequalities are problems expressed in the form of linear equations with the help of symbols like <, >, ≤, and ≥, and these are inequality symbols. In equations that represent inequality, the L.H.S and R.H.S of the equation are not equal; instead, they are either smaller than, greater than, smaller than, and equal, greater, or equal to each other. Usually, inequality equations can be solved using simple addition and subtraction methods. However, certain rules need to be followed when doing such calculations.

Inequality Equations

In algebra problems, there are many situations where an equation is not entirely balanced on both sides In other words, such equations, where the R.H.S. of the equation is not equal to the L.H.S, the equation is called an inequality equation.

When such expressions are written in the form of linear equations, then the equations are called linear inequalities. If the value of an equation component is not a real number, then a situation of inequality may arise in a linear equation.

In a linear equation, several symbols are used to represent inequality.

Symbols of Inequality in a Linear Equation

In a linear equation with inequality, a particular symbol is used to identify a particular type of inequality. The symbols <, >, ≤, and ≥ represent inequality in a linear equation, and every symbol represents a different condition of inequality.

The following are the examples that explain the meaning of each inequality symbol. 

4x + 7y < 85,

The inequality sign is < (less than) in the equation above. Thus 4x + 7y is less than 85, or the left-hand side of the equation will always be less than the right-hand side of the equation.

It is important to note here that, for any value of the variables in the equation, the total value of the left-hand side will always be less than 85.

We now look at a different example:

7x + 4y > 85,

The inequality sign > (greater than) has been used in this equation. It means that 7x + 4y will always be greater than 85, or the left-hand side of the equation will always be greater than that of the right-hand side of the equation. 

For this equation, the value of the left-hand side will always be greater than 85.

Let us now look at a third example:

10x ≤ 130,

In the above equation, the sign used to describe a condition of inequality is ≤, it is the sign for the condition of less than or equal to.

For this equation, the left-hand side component, for any value of the x, will remain less than or equal to 130 and will never be greater than the value on the right-hand side of equation 130.

We look at another such example:

10x ≥ 130,

The inequality sign used in the above equation is ≥, representing a greater or equal condition. It means that the value of the left-hand side of the equation will always be greater or equal to the right-hand side of the equation but will never be less than that of the values of the right-hand side of the equation.

Solving Inequalities with Addition.

Linear inequality equations can be solved by simple algebra calculations, including addition and subtraction.

Let us look at an example to understand the process of solving inequality in addition.

5x – 3 > 22,

To solve an equation with inequality with addition, we have to do addition on both sides of the equation, so the changes in the equation remain balanced.

Adding +3 in both sides of the equation, we get

5x – 3 + 3 > 22 + 3,

5x > 25,

Now, dividing the equation with 5 on both sides, we get

x >5.

Solving Inequalities With Subtraction

Linear inequalities in an equation can be solved by subtracting values on each side. 

Let us take another example:

3x + 9 < 10,

To solve this equation, let us subtract 9 from both sides of the equation:

We get,

3x + 9 – 9 < 10 – 9,

3x < 1,

Finally, we divide the equation with 3 into both sides:

x < 1/3.

Change in the Symbols of Inequality

While solving equations with inequality by addition and subtraction, sometimes equations may contain negative values on either side. When the negative or positive signs are removed or eliminated from either side of an equation, the inequality sign is also reversed,

Let us understand with the help of an example:

11x – 7y < -79,

Upon eliminating the negative sign from the equation:

-11x + 7y > 79,

Or, 

7y – 11x > 79.

Solved Examples

Example 1: Solve the inequality equation by addition.

15y – 18 > 32.

Solution: The given equation is

15y – 18 > 32

Adding +18 on both sides of the equation

15y – 18 + 18 > 32 + 18,

We get,

15y > 50,

Now, dividing 15 into both sides of the equation:

y > 10/3.

Example 2: Solve the inequality equation by subtraction.

25x + 16 > 36.

Solution: The given equation is

25x + 16 > 36

Subtracting both sides of the equation with 16, we get

25x + 16 – 16 > 36 – 16,

25x > 20,

Dividing both sides of the equation with 25, we get

x > 20/25,

Or

x > 4/5.

Example 3: Solve the following inequality equation.

5x + 5 < 15 + 10.

Solution: The given equation is:

5x + 5< 15 + 10,

First, add the values on the right-hand side of the equation without making changes to the other side:

5x + 5 < 25,

Now, subtracting 5 in both sides of the equation:

5x + 5 – 5 < 25 – 5,

5x < 20,

Finally, dividing both sides with 5:

5x/5 < 20/5,

Or

x < 4.

Conclusion

The symbols representing linear inequality equations are, >, <, ≤, and ≥. The symbols also help define and determine the conditions of inequality. We can solve linear inequality equations with subtraction and addition. Certain measures need to be followed, like the inequality symbol always being reversed, when they sign-in, an inequality equation changes from positive to negative and vice versa.