# Doubt solving a polynomial inequality

• greg_rack
In summary, the function is positive if and only if x<0, and it is not possible to solve the inequality for x>2.
greg_rack
Gold Member
Homework Statement
$$\sqrt{x^{2}-2x}-x+1>0$$
Relevant Equations
none
I got this function in a function analysis and got confused on how to solve its positivity;
I rewrote it as:
$$\sqrt{x^{2}-2x}>x-1 \rightarrow x^2-2x>x^2-2x+1$$
And therefore concluded it must've been impossible... but I'm certainly missing something stupid, since plotting the graphs of the two functions(##\sqrt{x^{2}-2x}## and ##x-1##) I see that the first is greater only for ##x<0##.

Maybe the flaw comes when I'm squaring both factors in the inequality... should I put the first factor in an absolute value, since it must be positive, as a square root?
But then, how do I solve it with the abs.?
I can deduce that the left term must be greater until the second term is negative... and for positive values it's the reversed situation, and this thinking works, but is there a more "rigid" procedure?

You need to study the various part this inequality have.

First, x<0 or x>2 is a condition to the root.

So in the case x<0 the inequality is immediate, since the root is always greater than zero, and x-1 is zero

Now you apply for x>2 too.

Take the union.

greg_rack
Your mistake is in the squaring, squaring inequalities just doesn't work sometimes. For example take the inequality -2<1 which is true. If you square it however you get the inequality 4<1 which is false..

Delta2 said:
Your mistake is in the squaring, squaring inequalities just doesn't work sometimes. For example take the inequality -2<1 which is true. If you square it however you get the inequality 4<1 which is false..
So, generally, you don't square inequalities? Is there a rule to avoid this type of mistakes?

I think the rule about squaring inequalities is that:

if both sides are positive then you can square it and the inequality keeps its direction

if both sides are negative then the inequality reverses direction after the squaring,

and if one side is negative and one positive then squaring is unpredictable!

greg_rack

## What is a polynomial inequality?

A polynomial inequality is an inequality that involves one or more variables raised to different powers, such as x^2, x^3, etc. The inequality can have terms that are constants, variables, or a combination of both.

## How do you solve a polynomial inequality?

To solve a polynomial inequality, you need to follow these steps:
1. Simplify the inequality by combining like terms and moving all terms to one side.
2. Factor the polynomial on the other side of the inequality.
3. Find the critical values by setting each factor equal to zero and solving for the variable.
4. Use a sign chart to determine the intervals where the polynomial is positive or negative.
5. Test a value from each interval in the original inequality to determine which intervals satisfy the inequality.

## What is the difference between solving a polynomial equation and solving a polynomial inequality?

The main difference between solving a polynomial equation and solving a polynomial inequality is the presence of an inequality symbol. When solving an equation, the goal is to find the value(s) of the variable that make the equation true. When solving an inequality, the goal is to find the range of values that make the inequality true.

## What are some common mistakes when solving a polynomial inequality?

Some common mistakes when solving a polynomial inequality include:
- Forgetting to flip the inequality sign when multiplying or dividing by a negative number.
- Not checking the critical values in the original inequality.
- Misinterpreting the sign chart and choosing the wrong intervals.
- Not simplifying the polynomial before factoring.
- Forgetting to include the equal sign when the inequality is "greater than or equal to" or "less than or equal to".

## What are some tips for solving a polynomial inequality?

Here are some tips for solving a polynomial inequality:
- Always simplify the polynomial before factoring.
- Be careful when multiplying or dividing by negative numbers.
- Check your critical values in the original inequality.
- Use a sign chart to help visualize the intervals.
- Test a value from each interval in the original inequality to confirm your solution.

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