Solve x²-4/9: Finding Where Function Values Are Negative

Q: Why does the parabola go negative between the roots?

Since $ x^2 - \frac{4}{9} $ is a parabola opening upward, it dips below the x-axis between its two zeros. The function equals zero at $ x = ±\frac{2}{3} $ and is negative in between.

Q: How do I know which direction the inequality goes?

When you have \( x^2 , think: "x-squared is less than this positive number." This means x must be close to zero, giving you the interval between the square roots.

Q: What if I got the wrong interval direction?

Always test a point! Pick $ x = 0 $: does \( 0^2 - \frac{4}{9} ? Yes, so zero should be in your solution interval. This confirms \( -\frac{2}{3} .

Q: Can I solve this by factoring instead?

Yes! Factor as \( (x - \frac{2}{3})(x + \frac{2}{3}) . For a product to be negative, the factors must have opposite signs, which happens between the zeros.

Q: Why not just graph this function?

Graphing is great for visualization! You'd see the parabola dips below the x-axis between $ x = -\frac{2}{3} $ and $ x = \frac{2}{3} $. Algebraic solving gives you the exact answer.

Quadratic Inequalities with Negative Function Values

Find the positive and negative domains of the function below:

$y=x^2-\frac{4}{9}$

Determine for which values of $x$ the following is true:

$f(x) < 0$

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Find the positive and negative domains of the function below:

$y=x^2-\frac{4}{9}$

Determine for which values of $x$ the following is true:

$f(x) < 0$

Step-by-step solution

To find where the function $y = x^2 - \frac{4}{9}$ is negative, solve the inequality:

$x^2 - \frac{4}{9} < 0$

This inequality simplifies to:

$x^2 < \frac{4}{9}$

Next, take the square root of both sides. Note the square root applies to both positive and negative values:

$-\frac{2}{3} < x < \frac{2}{3}$

Therefore, the values of $x$ for which the function $y = x^2 - \frac{4}{9}$ is negative are within the interval:

$-\frac{2}{3} < x < \frac{2}{3}$

Thus, the solution is correctly given by choice 2, with the interval $x$ such that:

$-\frac{2}{3} < x < \frac{2}{3}$

Final Answer

$-\frac{2}{3} < x< \frac{2}{3}$

Key Points to Remember

Essential concepts to master this topic

Rule: Set function less than zero and solve inequality
Technique: Find zeros first: $x^2 - \frac{4}{9} = 0$ gives $x = ±\frac{2}{3}$
Check: Test a value inside interval: $0^2 - \frac{4}{9} = -\frac{4}{9} < 0$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting both positive and negative square roots
Don't solve $x^2 < \frac{4}{9}$ as just $x < \frac{2}{3}$ = missing half the solution! The square root inequality gives both directions. Always write $-\frac{2}{3} < x < \frac{2}{3}$ to capture the complete interval.

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 does the parabola go negative between the roots?

Since $x^2 - \frac{4}{9}$ is a parabola opening upward, it dips below the x-axis between its two zeros. The function equals zero at $x = ±\frac{2}{3}$ and is negative in between.

How do I know which direction the inequality goes?

When you have $x^2 < \frac{4}{9}$ , think: "x-squared is less than this positive number." This means x must be close to zero, giving you the interval between the square roots.

What if I got the wrong interval direction?

Always test a point! Pick $x = 0$ : does $0^2 - \frac{4}{9} < 0$ ? Yes, so zero should be in your solution interval. This confirms $-\frac{2}{3} < x < \frac{2}{3}$ .

Can I solve this by factoring instead?

Yes! Factor as $(x - \frac{2}{3})(x + \frac{2}{3}) < 0$ . For a product to be negative, the factors must have opposite signs, which happens between the zeros.

Why not just graph this function?

Graphing is great for visualization! You'd see the parabola dips below the x-axis between $x = -\frac{2}{3}$ and $x = \frac{2}{3}$ . Algebraic solving gives you the exact answer.

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