Determine X for the Inequality: f(x) = x² > 0

Q: Why isn't x = 0 included in the solution?

Because we need $ f(x) > 0 $, which means strictly greater than zero. At x = 0, we get $ 0^2 = 0 $, which equals zero but is not greater than zero.

Q: What would happen if the inequality was f(x) ≥ 0 instead?

Then the answer would be all real numbers because $ x^2 \geq 0 $ for every x value, including x = 0 where $ x^2 = 0 $.

Q: How do I know that x² is always positive for non-zero values?

Any real number multiplied by itself gives a positive result. For example: $ (-3)^2 = (-3) \times (-3) = 9 > 0 $ and $ 5^2 = 5 \times 5 = 25 > 0 $.

Q: What does the notation x ≠ 0 mean exactly?

It means all real numbers except zero. This includes all positive numbers (1, 2, 3.5, 100...) and all negative numbers (-1, -2, -0.5, -100...), but excludes zero itself.

Q: Can I write the answer as x 0?

No! That would exclude either all negative or all positive numbers. The correct answer includes both positive and negative numbers, so we write $ x \neq 0 $.

Quadratic Inequalities with Sign Analysis

Given the function:

$y=x^2$

Determine for which values of x is $f\left(x\right) > 0$ true

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Given the function:

$y=x^2$

Determine for which values of x is $f\left(x\right) > 0$ true

Step-by-step solution

To solve this problem, let's follow our planned approach:

Consider the function $y = x^2$ .
The quadratic function $y = x^2$ is non-negative for all real numbers.
Specifically, $y = x^2 = 0$ only at $x = 0$ . Therefore, $y > 0$ when $x \neq 0$ .

Therefore, for $f(x) = x^2$ to be greater than zero, the condition is that $x \neq 0$ .

Thus, the solution to the problem is $x \neq 0$ .

Final Answer

$x\ne0$

Key Points to Remember

Essential concepts to master this topic

Rule: $x^2 \geq 0$ for all real numbers, equals zero only at x = 0
Technique: Test values: $(-2)^2 = 4 > 0$ and $(3)^2 = 9 > 0$
Check: Verify boundary: $0^2 = 0$ (not greater than 0) ✓

Common Mistakes

Avoid these frequent errors

Assuming x² > 0 for all real numbers including zero
Don't include x = 0 in your solution = wrong answer! At x = 0, we get 0² = 0, which is not greater than zero. Always exclude the value where the function equals zero when solving strict inequalities.

Practice Quiz

Test your knowledge with interactive questions

The graph of the function below intersects the X-axis at points A and B.

The vertex of the parabola is marked at point C.

Find all values of $$ x $$ where $f\left(x\right) > 0$ .

FAQ

Everything you need to know about this question

Why isn't x = 0 included in the solution?

Because we need $f(x) > 0$ , which means strictly greater than zero. At x = 0, we get $0^2 = 0$ , which equals zero but is not greater than zero.

What would happen if the inequality was f(x) ≥ 0 instead?

Then the answer would be all real numbers because $x^2 \geq 0$ for every x value, including x = 0 where $x^2 = 0$ .

How do I know that x² is always positive for non-zero values?

Any real number multiplied by itself gives a positive result. For example: $(-3)^2 = (-3) \times (-3) = 9 > 0$ and $5^2 = 5 \times 5 = 25 > 0$ .

What does the notation x ≠ 0 mean exactly?

It means all real numbers except zero. This includes all positive numbers (1, 2, 3.5, 100...) and all negative numbers (-1, -2, -0.5, -100...), but excludes zero itself.

Can I write the answer as x < 0 or x > 0?

No! That would exclude either all negative or all positive numbers. The correct answer includes both positive and negative numbers, so we write $x \neq 0$ .

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