Linear Function Sign Analysis: Finding Positive Regions on Coordinate Plane

Linear Function Analysis with X-Intercept Sign Rules

Look at the function shown in the figure.

When is the function positive?

xy-4-7

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 When is the function positive?
00:03 The function is positive when it's above the X-axis
00:07 and negative when the function is below the X-axis
00:16 Let's identify when the function intersects the X-axis
00:23 We'll identify when the function is positive and when it's negative
00:27 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Look at the function shown in the figure.

When is the function positive?

xy-4-7

2

Step-by-step solution

The function we see is a decreasing function,

Because as X increases, the value of Y decreases, creating the slope of the function.

We know that this function intersects the X-axis at the point x=-4

Therefore, we can understand that up to -4, the values of Y are greater than 0, and after -4, the values of Y are less than zero.

Therefore, the function will be positive only when

X < -4

 

3

Final Answer

4>x -4 > x

Key Points to Remember

Essential concepts to master this topic
  • Sign Rule: Function is positive where graph lies above x-axis
  • X-Intercept: At x = -4, function equals zero and changes sign
  • Check: For x < -4, pick x = -5: graph shows y > 0 ✓

Common Mistakes

Avoid these frequent errors
  • Confusing x and y values when reading intercepts
    Don't read the y-intercept (-7) as the x-intercept = wrong boundary! This gives x > -7 instead of x < -4. Always identify where the line crosses the x-axis to find the correct boundary.

Practice Quiz

Test your knowledge with interactive questions

Look at the function shown in the figure.

When is the function positive?

xy-4-7

FAQ

Everything you need to know about this question

How do I know which side of the x-intercept makes the function positive?

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Look at the graph direction! Since this line slopes downward (decreasing), the function is positive to the left of the x-intercept at x = -4.

Why is the answer x < -4 and not x > -4?

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The line crosses the x-axis at x = -4. For a decreasing function, values are positive before the intercept and negative after. So x < -4 gives positive y-values.

What does it mean for a function to be positive?

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A function is positive when its y-values are greater than zero. On a graph, this means the line is above the x-axis.

How can I verify my answer without the graph?

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Pick a test point! Choose x = -5 (which is < -4). If the function gives a positive y-value at x = -5, then x<4 x < -4 is correct.

Does the direction of the inequality matter?

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Yes! For decreasing functions, positive regions are typically where x is less than the x-intercept. For increasing functions, it's usually where x is greater than the intercept.

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