### Linear inequalities with radicals

**Linear inequalities with radicals**

While dealing with inequalities, we also need to consider the **domain of radical expressions**. We take into consideration both the solutions, one which we get after solving inequality by squaring both sides and other by setting radicand ≥ 0. lets work on some examples to understand this process better.

Example1. Solve the following inequality and write the solution in interval form.

Solution:

Squaring both sides,

x – 1 ≤ 9

x ≤ 10 ——(i)

Since radicals are not defined for negative values so radicand must be positive. Therefore, x-1 ≥ 0

=> x ≥ 1 ——–(ii)

Combining the solutions (i) and (ii), we get final solution as,

1 ≤ x ≤ 10 => [1 ,10]

Example2. Solve the following inequality and write the solution in interval form.

Solution: Also,

* x ≥ 3
*

*(multiplied with -1 and inequality sign changed)*

squaring both sides,

x -3 ≤ 36

x ≤ 39

combining two solutions we get, 3 ≤ x ≤ 39

Final answer in interval form [3,39]

Example3.Solve the following inequality and write the solution in interval form.

Solution: We can rewrite the given inequality as,

Since LCM of denominators 2 and 3 is 6 so we multiply exponents of both sides with 6 and get,

=>

As can never be negative so x-1 ≤ 0

x≤ 1 —— (i)

Also , being an even radical is never defined for negative values which restrict the given solution to x ≥ 0 —— (ii)

So final solution, after combining (i) and (ii) is given as,

0≤x≤1 => [0,1]

Example4. Solve the inequality

Solution: Rewriting the given inequality as,

Since lcm of denominators 2 and 4 is 4, So raising exponents on both sides by 4 and we get,

x ( x-16) ≥ 0

As we know that even radicals are not defined for negative values which restrict domain to x ≥ 0,

Also, x-16≥0 => x≥ 16

So final solution is given as, [16,∞)

**Practice problems:**

Solve the following inequalities and write the answer in interval form.

Answers:

- (0,1)
- [1/2, 5]
- [32,∞)
- ( -3/11 , 2)