Inequalities-Solving Inequality Worksheets Using Multiplication and Division

Inequalities in algebra are those equations expressed with symbols like <, >, ≤, and ≥.  These are inequality symbols. In inequality equations, the right and left-hand sides of the equation are not equal; instead, they are either greater than, smaller than, greater or equal, and smaller or equal to each other. Simple inequalities can be solved by using simple calculations like multiplication and division. However, there are certain rules and methods for these calculations.

Inequalities

There are certain situations when an equation is not completely satisfied in mathematical problems. In simple terms, in such equations, the L.H.S. of the equation doesn’t equal the R.H.S. of the equation; these equations are called inequality equations.

When such scenarios are expressed in a linear equation, the equations are called linear inequalities. Inequalities may arise when the value of an equation component is not an actual number. Various inequality symbols are used to represent inequalities in an equation.

Symbols of Inequality

In an equation representing an expression of inequality, a particular symbol identifies a certain kind of inequality. The symbols used to represent inequality of an equation are called inequality symbols; they are, <, >, ≤ and ≥.

Each of these symbols represents a different kind of inequality.

Let us understand the meaning of these signs with examples.

2x + 3y < 100

In the given equation, we can see an inequality sign in the equation < (less than). It indicates 2x + 3y is less than 100, or the L.H.S. of the equation is less than the R.H.S. of the equation.

This means that for any value of variable x and y, the composite value of the L.H.S. of the given equation will always be less than 100.

Now, let us look at a different example:

3x + 2y > 100,

We have used the inequality sign > (greater than) in the above equation. Therefore 3x + 2y is greater than 100, or the L.H.S. of the equation is greater than the R.H.S. of the equation. 

Here, the value of the L.H.S. will always be greater than 100.

Now, consider a third scenario:

5x ≤ 50,

The sign of inequality in the above equation is ≤, that is of ‘less than or equals to’.

This means that the 5x for any value of the x will be less or equal to 50.

Similarly, for an equation

5x ≥ 50,

The sign of inequality (≥) is greater or equal to, representing that for any value of the variable, the value of L.H.S. will always be greater or equal to R.H.S., but will never be less than that of R.H.S.

Solving Inequalities with Multiplication.

Simple inequality equations can be solved by simple calculations, such as addition, subtraction, multiplication, and division.

Let us understand the process of solving inequality with multiplication with an example:

x/7 < 21,

To solve this equality with multiplication, we have to perform multiplication on both sides of the equation, so the changes in the equation remain in the right balance.

Now, multiplying 7 in both sides of the equation, we get,

7. (x/7) < 21.(7),

x < 147.

 Solving Inequalities With Division

Similar to multiplication, inequalities can be solved by the division method also. Let us look at another example for the same:

50x ≤ 100,

To solve this equation, let us divide 50 from both sides of the equation.

We get, 50x/50 ≤ 100/50,

Finally, x ≤ 100.

Changing Symbols of Inequality

In some problems, while solving inequality equations with division and multiplication, we may have negative values on either side of the equation. When we eliminate the negative or positive sign from any side of the equation, the inequality sign is also changed.

Let us understand with the help of an example:

6x – 7y < -49,

To eliminate the negative sign from the R.H.S of the equation, we multiply both sides with -1,

-6x + 7y > 49,

Or, 

7y – 6x > 49

Solved Examples

Example 1: Solve the inequality equation by multiplication.

x/10 > 3/5

Solution: The given equation is:

x/10 > 3/5,

Multiplying, 10 in both sides of the equation:

10.(x/10) > (3/5).10,

We get,

x > 6.

Example 2: Solve the Inequality equation by division.

25y ≤ 500

Solution: The given equation is:

25y ≤ 500,

Dividing both sides of the equation with 25, we get

25y/25 ≤ 500/25,

y ≤ 20.

Example 3: Solve the following inequality by multiplication and division.

5x/2 < 15

Solution: The given equation is:

5x/2 < 15,

First, multiplying the equation with 2:

2.(5x/2) < 15.(2),

5x < 30,

Now, dividing the equation with 5:

5x/5 < 30/5,

x < 6

Conclusion

Inequality symbols represent the inequality equations. The inequality symbols are >, <, ≤, and ≥. 

The simple inequalities can be solved by simple algebra, like multiplication and division. However, these calculations follow certain rules. The inequality symbol is reversed as the sign in an inequality equation changes from positive to negative and vice versa.