Solving Square Root Inequalities

Solving square root inequalities involves the process of solving square root equation with inequalities in detail. The square root can be easily solved by performing squaring operation for the given equation. The comparison process is carried out with the help of inequality sign. So the comparison operation carried out in the square root equation is termed as solving square root equation. The following are the example problems for square root inequalities for solving.

 

Square Root Inequalities Example Problems:

 

Example 1:

Solve the inequalities with square roots.

` sqrt(w^2-13w + 52) <= 4` 

Solution:

Given equation is
` sqrt(w^2-13w + 52) <= 4` 

To neglect square root from the above equation perform squaring operation on both sides

`[sqrt(w^2-13w + 52)]^2 <= (4)^2`

And simplify. 
` w^2-13w+52<= 16`

Make the above equation in factor form.
` w^2- 13 w + 36 <= 0`

By solving above quadratic equation, it results two solutions
`w <= 4` and `w <= 9 ` is the answer.

 

 

Example 2:

Solve the inequalities with square roots.

`sqrt(2w + 3) >= 5` 

Solution:

Given equation is
`sqrt(2w + 3) >= 5`

To neglect square root from the above equation perform squaring operation on both sides
`[sqrt(2w + 3)]^2 >= 5^2`

And simplify 
`2w + 3 >= 25`

Solve for w. 
 `2w >= 22` 

`w >= 11` is the answer.

 

Example 3:

Solve the inequalities with square roots.

`sqrt(w + 7) <= w - 5` 

Solution:

Given equation is
`sqrt(w + 7) <= w - 5`

To neglect square root from the above equation perform squaring operation on both sides
`[sqrt(w + 7)]^2 <= (w - 5)^2`

Simplify the above equation 
`w + 7 <= w^2- 10 w + 25`

Change the above equation in factor form. 
`w^2- 11 w + 18 <= 0`

By solving above quadratic equation, it results two solutions 
`w <= 2` and `w <= 9` is the answer.

 

Square Root Inequalities Practice Problems:

 

1) Solve the inequalities with square roots.

 ` sqrt(w-1) <= w - 3`

Answer: `w <= 5` `and` `w<= 2`

2) Solve the inequalities with square roots.

     ` sqrt(w^2-12w+32) >= 0` 

Answer:` w >= 4 and w >= 8 `