 Again, solving inequalities is more important than solving equalities, so let's see if we can find the solutions to inequalities involving roots. Remember that the process of solving inequalities begins by ignoring the inequality and solving the corresponding equality for the critical values. But then, like a good human being, we have to go back and address the inequality. And mathematically, the critical values partition the number line into several intervals and we test points at each interval to see where the inequality is satisfied. And so this approach works with linear and quadratic inequalities, but now we have a new type of critical value, because the square root of x is undefined if x is less than zero, a radical inequality includes a second inequality in itself. In particular, we have to make sure that the thing that we're taking the square root of is not negative. And the critical values for this inequality are also critical values for the original. For example, let's solve square root for x minus seven greater than or equal to five. So first, we'll turn a blind eye and pretend that this is an equality and solve the equation. So one critical value is x equals eight. Now we do have an inequality, so like a good math student or a good human being, we have to acknowledge that inequality and we'll check to see if x equals eight satisfies this inequality. And the easy way to do that is to remember that x equals eight satisfies the equation and our inequality allows for equality. So x equals eight does satisfy the inequality and so we should include it. Now there's another important feature here. Since the equation involves a square root, the rad account must be non-negative. In other words, the thing under the square root for x minus seven must be greater than or equal to zero. And so this is a second inequality that's embedded within this original. Well, it's another inequality so we should solve it and find the critical points. And that gives us a critical value of seven fourths. Again, if we hope to be a good math student or a good human being, we have to acknowledge that we started with an inequality. If x equals seven fourths, if we try to substitute that back into our original inequality, we get a false statement. So we have to exclude this point and we'll use an open circle. And so notice we now have three intervals to check. So in the first interval we could check x equals zero. Now this gives a square root of negative seven and since this isn't even defined, x equals zero is not a solution so we should exclude this first interval. In the next interval we could check a point in the middle, say, I don't know, x equals two, and we find that this is false so we need to exclude this interval. In this last interval we could go big because this includes everything to the right of eight So we'll check x equals one million and we find this is true so we should include this last interval and so our graph of the solution looks like this. And from the graph of the solution we can write down our answer in interval notation x is going to be the interval from eight included all the way after. As another example, square root of x squared minus x minus twenty greater than or equal to x minus two so we'll start by solving the equality since this is the form root equals, we'll square both sides to get rid of the root, we'll expand the right hand side and collect like terms and solve the equation. This gives us a critical value of eight which we test and since the inequality is true then we include x equals eight because this is a square root we need to make sure that the radicand is non-negative so we need x squared minus x minus twenty to be greater than or equal to zero so we'll ignore the inequality and solve giving us solutions x equals five or x equals negative four now ordinarily we check these critical values against the inequality we were solving but remember the inequality we're actually solving is this one so we'll check our critical values to see whether they solve this original inequality and we find that we should include negative four and exclude five the three critical values give us four intervals to test so we'll test a point in each one over on the left we can go big if x equals negative one million which is true since square root is a positive number and what's on the right is a negative number so we include this interval the next interval includes zero so we can test it out if x equals zero we have the square root of a negative number and so radical is undefined so we exclude this interval so in the next interval we can try a test point of x equals six and this is false so we'll have to exclude this interval and in our last interval we can try x equals one million which is true so we should include this interval in our solution set now since we have the graph of our solution set we can write our answer in interval notation