Solve the Quadratic Inequality: -x² - 9 > 0

Q: Why can't we solve x² < -9 like a regular inequality?

Because squares are never negative! Any real number squared gives a result ≥ 0. Since -9 is negative, there's no real number x where x² could be less than -9.

Q: What does 'no solution' actually mean?

It means there are no real numbers that make the inequality true. The solution set is empty - we write this as ∅ or { } in set notation.

Q: Could I have made an algebra mistake somewhere?

Double-check your work! The steps are: $ -x^2 - 9 > 0 $ → $ -x^2 > 9 $ → \( x^2 . Remember to flip the inequality when multiplying by -1.

Q: Are there any numbers that work in complex numbers?

Yes! In complex numbers, x = ±3i would work, but this problem asks for real solutions only. In algebra class, we typically work with real numbers unless specified otherwise.

Q: How do I recognize when an inequality has no solution?

When you get \( x^2 When you get something impossible like \( 5 When the parabola doesn't cross the x-axis in the required direction

Q: What's the difference between no solution and x = 0?

No solution means no value of x works at all. x = 0 means zero is the only solution. These are completely different! Always substitute to check which case you have.

Quadratic Inequalities with No Solution

Solve the following equation:

$-x^2-9>0$

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Understand the problem

Solve the following equation:

$-x^2-9>0$

Step-by-step solution

To solve this quadratic inequality, $-x^2 - 9 > 0$ , we will follow these steps:

Step 1: Identify the quadratic expression $-x^2 - 9$ .
Step 2: Attempt transformation and determine when the expression $-x^2 - 9$ , can be greater than zero.

Let's analyze the equation:

Rewrite the inequality:
$-x^2 - 9 > 0$

Add 9 to both sides:
$-x^2 > 9$

Multiply the entire inequality by $-1$ and remember to reverse the inequality sign:
$x^2 < -9$

Observe the inequality $x^2 < -9$ :
Note that $x^2$ , being a square of any real number, is always greater than or equal to zero.

As $x^2$ cannot be less than negative nine for any real number $x$ , the inequality has no solution in the realm of real numbers.

Therefore, the correct answer is:

There is no solution.

Final Answer

There is no solution.

Key Points to Remember

Essential concepts to master this topic

Rule: Squares of real numbers are always non-negative
Technique: Rearrange $-x^2 - 9 > 0$ to get $x^2 < -9$
Check: Since $x^2 \geq 0$ but $-9 < 0$ , no solution exists ✓

Common Mistakes

Avoid these frequent errors

Trying to take square roots of both sides
Don't try to solve $x^2 < -9$ by taking square roots = undefined operation! You cannot take the square root of a negative number in real numbers. Always recognize when $x^2$ is compared to a negative value - this means no real solution exists.

Practice Quiz

Test your knowledge with interactive questions

Solve the following equation:

$$ x^2+4>0 $$

FAQ

Everything you need to know about this question

Why can't we solve x² < -9 like a regular inequality?

Because squares are never negative! Any real number squared gives a result ≥ 0. Since -9 is negative, there's no real number x where x² could be less than -9.

What does 'no solution' actually mean?

It means there are no real numbers that make the inequality true. The solution set is empty - we write this as ∅ or { } in set notation.

Could I have made an algebra mistake somewhere?

Double-check your work! The steps are: $-x^2 - 9 > 0$ → $-x^2 > 9$ → $x^2 < -9$ . Remember to flip the inequality when multiplying by -1.

Are there any numbers that work in complex numbers?

Yes! In complex numbers, x = ±3i would work, but this problem asks for real solutions only. In algebra class, we typically work with real numbers unless specified otherwise.

How do I recognize when an inequality has no solution?

When you get $x^2 < \text{negative number}$
When you get something impossible like $5 < 3$
When the parabola doesn't cross the x-axis in the required direction

What's the difference between no solution and x = 0?

No solution means no value of x works at all. x = 0 means zero is the only solution. These are completely different! Always substitute to check which case you have.

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