Solve the Inequality: Find X Where f(x) < 0

Q: How do I know which side of the x-axis is negative?

The x-axis represents y = 0. When a graph is below the x-axis, the y-values (function values) are negative. When it's above, they're positive.

Q: What if the graph touches but doesn't cross the x-axis?

If the graph only touches the x-axis at a point, that point has f(x) = 0, not f(x)

Q: Why do I write the answer as 'x 8' instead of one interval?

Because the function is negative in two separate regions that don't connect! The graph is positive between x = 2 and x = 8, so you need two different intervals.

Q: How can I double-check my answer without the graph?

Pick test points in each interval! Try x = 0 (should give negative), x = 5 (should give positive), and x = 10 (should give negative) to verify your regions.

Q: What's the difference between f(x) < 0 and f(x) ≤ 0?

f(x) strictly below the x-axis (doesn't include the x-intercepts). f(x) ≤ 0 would include the points where the graph touches or crosses the x-axis too.

Graph Analysis with Inequality Intervals

Based on the data in the diagram, find for which X values the graph of the function $f\left(x\right) < 0$

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Based on the data in the diagram, find for which X values the graph of the function $f\left(x\right) < 0$

Step-by-step solution

To solve for when $f(x) < 0$ on the graph, we follow these steps:

Step 1: Locate the x-intercepts, where the curve intersects the x-axis. These intercepts are $x = 2$ and $x = 8$ .
Step 2: Analyze the sections determined by these intercepts. The graph is below the x-axis to the left of $x = 2$ and to the right of $x = 8$ .

By visually inspecting the graph, it is evident that:

The function $f(x)$ is below the x-axis (i.e., negative) for $x < 2$ and $x > 8$ .

Therefore, the solution to the problem is that the graph of the function is negative for $x > 8$ or $x < 2$ .

Final Answer

$x > 8$ or $x < 2$

Key Points to Remember

Essential concepts to master this topic

Rule: Function is negative when graph lies below x-axis
Technique: Find x-intercepts at x = 2 and x = 8, then check regions
Check: Pick test points: f(0) < 0 and f(10) < 0 confirm answer ✓

Common Mistakes

Avoid these frequent errors

Reading where function is positive instead of negative
Don't look at where the graph is above the x-axis when asked for f(x) < 0 = completely opposite answer! This gives you the wrong intervals. Always identify what the inequality symbol means: < 0 means below the x-axis.

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 side of the x-axis is negative?

The x-axis represents y = 0. When a graph is below the x-axis, the y-values (function values) are negative. When it's above, they're positive.

What if the graph touches but doesn't cross the x-axis?

If the graph only touches the x-axis at a point, that point has f(x) = 0, not f(x) < 0. Only include intervals where the graph is strictly below the axis.

Why do I write the answer as 'x < 2 or x > 8' instead of one interval?

Because the function is negative in two separate regions that don't connect! The graph is positive between x = 2 and x = 8, so you need two different intervals.

How can I double-check my answer without the graph?

Pick test points in each interval! Try x = 0 (should give negative), x = 5 (should give positive), and x = 10 (should give negative) to verify your regions.

What's the difference between f(x) < 0 and f(x) ≤ 0?

f(x) < 0 means strictly below the x-axis (doesn't include the x-intercepts). f(x) ≤ 0 would include the points where the graph touches or crosses the x-axis too.

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