SOLVING INEQUALITIES

We can solve simple inequalities as we are solving simple equations using the properties of equality.

That is, simple inequalities can be solved by adding, subtracting, multiplying or dividing both sides until the variable is isolated.    

Reversing the Sign of the Inequality

When we solve inequalities, sometimes, we may have to reverse the inequality symbol.  

We have to reverse the inequality sign in the following two cases.

Case 1 :

When we multiply or divide each side of an inequality by a negative number, we must reverse the inequality symbol to maintain a true statement.   

Case 2 :

When we interchange the quantities on the left side and right side of an inequality, we must reverse the inequality symbol to maintain a true statement.

Example 1 :

Solve the following inequality : 

5x - 4  ≥  4x + 6

Solution : 

Solve for x : 

5x - 4  ≥  4x + 6

Subtract 4x from each side. 

x - 4  ≥  6

Add 4 to each side. 

x  ≥  10

Example 2 :

Solve the following inequality : 

4z + 8  ≤  12

Solution : 

Solve for z : 

4z + 8  ≤  12

Subtract 8 from each side. 

4z  ≤  4

Divide each side by 4.  

z  ≤  1

Example 3 :

Solve the following inequality : 

-m + 3  ≥  -4m + 6

Solution : 

Solve for m : 

-m + 3  ≥  -4m + 6

Add 4m to each side. 

3m + 3  ≥  6

Subtract 3 from each side.  

3m  ≥  3

Divide each side by 3.

m  ≥  1

Example 4 :

Solve the following inequality : 

-x + 2  >  7

Solution : 

Solve for x : 

-x + 2  ≥  7

Subtract 2 from each side.  

-x  >  5

Multiply each side by -1.  

x  <  -5

Example 5 :

Solve the following inequality : 

2  >  3x - 10

Solution : 

Solve for x : 

2  >  3x - 10

Add 10 to each side.   

12  >  3x

Divide each side by 3.

4  >  x

Interchange the quantities on the left and right side.

x  <  4

Example 6 :

Solve the following inequality : 

16 - 2x  ≤  28

Solution : 

Solve for x : 

16 - 2x  ≤  28

Subtract 16 from each side.    

-2x  ≤  12

Divide each side by -2.

x  ≥  -6

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