Find Values Where f(x) > 0: Analyzing Function with Zero Points at -10, -6, and -2

Q: How do I know which intervals are positive without the actual function?

Use the graph provided! Look for where the curve is above the x-axis. You can also test points in each interval - pick any x-value between zeros and see if it gives a positive result.

Q: Why does the function change from positive to negative at each zero?

At each zero, the function crosses the x-axis. This means it goes from positive to negative (or vice versa) at each crossing point, assuming the zero has odd multiplicity.

Q: What if I can't tell from the graph which regions are positive?

Choose a test point in each interval! For example, test x = -12 (left of -10), x = -8 (between -10 and -6), x = -4 (between -6 and -2), and x = 0 (right of -2).

Q: Do I include the zero points in my answer?

No! The question asks for f(x) > 0, which means strictly greater than zero. Since f(x) = 0 at the zeros, those points are not included in the solution.

Polynomial Inequalities with Multiple Zero Points

Find all values of x

where $f\left(x\right) > 0$ .

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Step-by-step written solution

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Understand the problem

Find all values of x

where $f\left(x\right) > 0$ .

Step-by-step solution

We are given a problem involving the function $f(x)$ and asked to find the set of all $x$ such that $f(x) > 0$ . This implies finding those segments of the x-axis where the function is above the x-axis when graphed.

We can analyze the graph to solve the problem:

Firstly, we identify intersecting points on the x-axis (roots) from the graph directly. Let's assume the x-intercepts happen at $x = -6$ and $x = -2$ .
The quadratic nature suggests segments between and beyond these intercepts where $f(x) > 0$ .
Given it's upward-facing between $-10$ and $-6$ , and $-6$ to $-2$ , this evaluates that $f(x)$ is negative or flat at these technology-derived points.
Therefore, determining intervals requires examining external points:
The graph, based on inferences together, leads to positive $f(x) > 0$ for $x > -2$ or $x < -10$ , verified by factual plot exploration devices.

Therefore, the solution is that $x > -2$ or $x < -10$ .

Final Answer

$x > -2$ or $x > -10$

Key Points to Remember

Essential concepts to master this topic

Zero Analysis: Identify where function crosses x-axis at given points
Sign Testing: Check function behavior between zeros: f(-8) > 0, f(-4) < 0
Interval Check: Verify positive regions match graph behavior above x-axis ✓

Common Mistakes

Avoid these frequent errors

Assuming function is positive between all zeros
Don't assume f(x) > 0 between every pair of zeros = wrong intervals! Polynomials alternate signs at each zero crossing. Always test a point in each interval to determine the actual sign.

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

How do I know which intervals are positive without the actual function?

Use the graph provided! Look for where the curve is above the x-axis. You can also test points in each interval - pick any x-value between zeros and see if it gives a positive result.

Why does the function change from positive to negative at each zero?

At each zero, the function crosses the x-axis. This means it goes from positive to negative (or vice versa) at each crossing point, assuming the zero has odd multiplicity.

What if I can't tell from the graph which regions are positive?

Choose a test point in each interval! For example, test x = -12 (left of -10), x = -8 (between -10 and -6), x = -4 (between -6 and -2), and x = 0 (right of -2).

Do I include the zero points in my answer?

No! The question asks for f(x) > 0, which means strictly greater than zero. Since f(x) = 0 at the zeros, those points are not included in the solution.

How can I double-check my interval notation?

Pick one test point from each interval you think is positive and verify f(x) > 0 there. Also make sure your intervals don't include the boundary points where f(x) = 0.

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