Finding Negative Regions: When is f(x) < 0 on Linear Function Graph

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

Look at the graph position relative to the x-axis! The blue line is below the x-axis (negative) when $ x and above the x-axis (positive) when \( x > 2 $.

Q: What does the point (2, 0) mean for this problem?

The point (2, 0) is the x-intercept where the function equals zero, not negative or positive. It's the boundary point that separates the negative and positive regions of the function.

Q: Can I use algebra instead of just reading the graph?

Absolutely! If you have the equation $ f(x) = mx + b $, set it less than zero: \( mx + b , then solve for x. The graph method and algebraic method should give the same answer.

Q: Why is the answer x < 2 and not x ≤ 2?

Because we need $ f(x) (strictly less than zero). At \( x = 2 $, we have $ f(2) = 0 $, which doesn't satisfy the strict inequality.

Q: What if I can't see the exact x-intercept clearly?

Look for labeled points on the graph! Here, the point (2, 0) is marked where the line crosses the x-axis. You can also use the grid lines to estimate coordinates accurately.

Linear Functions with Negative Value Intervals

When is $f(x)<0$ ?

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Understand the problem

When is $f(x)<0$ ?

Step-by-step solution

To determine when $f(x) < 0$ , we analyze the graph shown. The point where the blue line intersects and crosses below the x-axis forms the critical point of interest.

By observing the graph, we see the blue line represents $f(x)$ . Visual interpretation shows that the blue line dips below the x-axis before it reaches the $x = 2$ point and moves up at exactly this point.

Therefore, since the blue graph representing $f(x)$ is below the x-axis when $x < 2$ , it implies that the interval for which $f(x) < 0$ holds true is precisely $x < 2$ .

Hence, the solution to this problem is $x < 2$ .

Final Answer

$x < 2$

Key Points to Remember

Essential concepts to master this topic

Rule: Function is negative when graph lies below x-axis
Technique: Find x-intercept where f(x) = 0, here at x = 2
Check: Test a point: when x = 0, f(0) = -8 < 0 ✓

Common Mistakes

Avoid these frequent errors

Confusing x-intercept with where function is negative
Don't think f(x) < 0 means x < 0 = wrong interval! The x-intercept at x = 2 divides positive and negative regions. Always identify which side of the x-intercept makes the function negative by checking the graph position relative to the x-axis.

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

How do I know which side of the x-intercept is negative?

Look at the graph position relative to the x-axis! The blue line is below the x-axis (negative) when $x < 2$ and above the x-axis (positive) when $x > 2$ .

What does the point (2, 0) mean for this problem?

The point (2, 0) is the x-intercept where the function equals zero, not negative or positive. It's the boundary point that separates the negative and positive regions of the function.

Can I use algebra instead of just reading the graph?

Absolutely! If you have the equation $f(x) = mx + b$ , set it less than zero: $mx + b < 0$ , then solve for x. The graph method and algebraic method should give the same answer.

Why is the answer x < 2 and not x ≤ 2?

Because we need $f(x) < 0$ (strictly less than zero). At $x = 2$ , we have $f(2) = 0$ , which doesn't satisfy the strict inequality.

What if I can't see the exact x-intercept clearly?

Look for labeled points on the graph! Here, the point (2, 0) is marked where the line crosses the x-axis. You can also use the grid lines to estimate coordinates accurately.

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