Solving Radical Inequalities

In this exercise, we'll solve a radical inequality. We are given two minus two x over three is greater than or equal to the square root of x plus four.

Solving radical inequalities requires two main considerations: identifying the domain for which the root is defined, and ensuring both sides are handled correctly when squaring.

So, x plus four must be greater than or equal to zero.

By subtracting four from both sides, our first condition is that x must be greater than or equal to negative four.

Now, notice that the square root is always non-negative. For the left side to be greater than or equal to it, the left side must also be non-negative.

We rearrange this to isolate x. Multiplying by three, we get six minus two x is greater than or equal to zero.

Then, six is greater than or equal to two x, which means x must be less than or equal to three.

With our domain and sign constraints established, we can safely square both sides of the original inequality.

Let's expand the left side using the binomial square formula.