It's not so much the square root -- it's the simple fact of squaring both sides of an inequality.Mr Davis 97 said:So I am trying to solve a simple rational inequality: ##\sqrt{x} < 2x##. Now, why can't I just square the inequality and go on my way solving what results? What precisely is the reason that I need to be careful when squaring the square root?
Squaring both sides of an equation or inequality is not a one-to-one operation, so isn't reversible. Once you have squared something, you can no longer determine whether the thing that was squared was positive or negative.Mr Davis 97 said:Well, the signs are not an issue in this scenario, but if I don't pay attention to the fact that I am squaring, I get an extraneous result. For example, if we go from ##\sqrt{x} < 2x \implies x < 4x^2 \implies x(4x - 1) > 0##, which then gives us ##x > \frac{1}{4}## or ##x < 0##. However, ##x < 0## is obviously extraneous because of the square root in the beginning of the problem, which implies that x must be positive. Thus, what is it about squaring that causes the extraneous solution set to occur?
So would a hard and fast rule be to just proceed as normal (i.e. square both sides and solve for x), and then after all is said done, check back to see whether you have an extraneous answers?Mark44 said:Squaring both sides of an equation or inequality is not a one-to-one operation, so isn't reversible. Once you have squared something, you can no longer determine whether the thing that was squared was positive or negative.
Leaving aside inequalities for the moment, consider ##\sqrt{x} = -2x##. Squaring both sides gives ##x = 4x^2## or x(4x - 1) = 0, with solutions x = 1/4 or x = 0. x = 1/4 is not a solution of the original equation. This extraneous solution arose because of the squaring operation.
Yes, this is something you should do. This would also include raising both sides to the fourth or other even power.Mr Davis 97 said:So would a hard and fast rule be to just proceed as normal (i.e. square both sides and solve for x), and then after all is said done, check back to see whether you have an extraneous answers?
An irrational inequality is an inequality that contains one or more irrational numbers, such as square roots. These numbers cannot be expressed as a ratio of two integers and have an infinite number of non-repeating decimal places.
The square root is important because it is an irrational number that often appears in irrational inequalities. To solve these types of inequalities, we need to isolate the variable with the square root and then square both sides of the inequality. This helps us to get rid of the square root and find the solution.
Yes, for example, let's solve the inequality √x + 2 < 5. First, we subtract 2 from both sides to isolate the square root: √x < 3. Then, we square both sides of the inequality: x < 9. So, the solution to this irrational inequality is x < 9.
Yes, there are a few special rules to keep in mind. First, when you square both sides of the inequality, you must consider both the positive and negative solutions. Second, you need to make sure that the original inequality is true for the solution you find. And lastly, if you multiply or divide by a negative number, you must flip the direction of the inequality sign.
Solving irrational inequalities is often used in fields such as engineering, physics, and economics to model and solve real-world problems. For example, it can be used to determine the range of values for a certain variable in a given situation or to find the optimal solution in a problem involving irrational numbers.