Given the function is negative in the domain $$ 4 > x $$ Find the equation of the line given that it passes through the point $$ (5,9) $$

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Positivity and Negativity Linear Function Practice Problems

Master finding domains of positivity and negativity for linear functions with step-by-step practice problems, inequality solving, and graph interpretation.

📚Practice Finding Positivity and Negativity Domains

Determine when linear functions are positive or negative using graphs
Solve inequalities to find domains of positivity and negativity
Identify x-intercepts and their relationship to function signs
Analyze linear functions above and below the x-axis
Apply algebraic methods to determine function positivity domains
Connect graphical and algebraic approaches for complete understanding

Understanding Positivity and Negativity

Complete explanation with examples

Positivity and Negativity of a Linear Function

The function is positive when it is above the $X$ axis when $Y<0$

The function is negative when it is below the $X$ axis as $Y>0$

When we are asked what the domains of positivity of the function are, we are actually being asked in which values of $X$ the function is positive: when it is above the $X$ axis.

In which values of $X$ does the function obtain positive $Y$ values?

When we are asked what the domain of negativity of the function is, we are actually being asked in which values of $X$ the function is negative: when it is below the $X$ axis.

In which values of $X$ does the function obtain negative $Y$ values?

1 - Positivity and Negativity of a Linear Function

Detailed explanation

Practice Positivity and Negativity

Test your knowledge with 4 quizzes

Choose the equation that represents a line with a negative domain of $$ 0 < x $$ .

Examples with solutions for Positivity and Negativity

Step-by-step solutions included

Exercise #1

Look at the linear function represented in the diagram.

When is the function positive?

Step-by-Step Solution

The function is positive when it is above the X-axis.

Let's note that the intersection point of the graph with the X-axis is:

$(2,0)$ meaning any number greater than 2:

$x > 2$

Answer:

$x>2$

Video Solution

Exercise #2

Look at the function shown in the figure.

When is the function positive?

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

Answer:

$-4 > x$

Video Solution

Exercise #3

Given the linear function of the drawing.

What is the negative domain of the function?

Step-by-Step Solution

The function is negative when it is below the Y-axis.

Note that the graph always remains above the X-axis, meaning it is always positive.

Answer:

The always positive function

Video Solution

Exercise #4

Look at the function graphed below.

What are the areas of positivity and negativity of the function?

Step-by-Step Solution

Let's remember that the function is positive when it is above the X-axis. The function is negative when it is below the X-axis.

Let's note that the intersection point of the graph with the X-axis is:

$(3.5,0)$

This means that when $x>3.5$ , it is below the X-axis and when $x < 3.5$ , it is above the X-axis.

In other words, the function is positive when $x < 3.5$ and the function is negative when $x>3.5$ .

Answer:

^{Positive $x<3.5$}

^{Negative $x>3.5$}

Video Solution

Exercise #5

Given the function of the graph.

What are the areas of positivity and negativity of the function?

Step-by-Step Solution

When we are asked what the domains of positivity of the function are, we are actually being asked at what values of X the function is positive: it is above the X-axis.

At what values of X does the function obtain positive Y values?

In the given graph, we observe that the function is above the X-axis before the point X=7, and below the line after this point. That is, the function is positive when X>7 and negative when X<7,

And this is the solution!

Answer:

^{Positive $7 > x$}

^{Negative $7 < x$}

Video Solution

Frequently Asked Questions

Everything you need to know about Positivity and Negativity

How do you find the domain of positivity of a linear function?

To find the domain of positivity, set the linear function greater than zero (y > 0) and solve the inequality. For example, with y = 4x - 2, solve 4x - 2 > 0 to get x > 0.5. This means the function is positive when x is greater than 0.5.

What is the difference between positive and negative domains of a function?

The positive domain shows where the function is above the x-axis (y > 0), while the negative domain shows where it's below the x-axis (y < 0). These domains are separated by the x-intercept where the function equals zero.

How do you solve linear function negativity problems step by step?

Follow these steps: 1) Set the function less than zero (y < 0), 2) Write the inequality using the function expression, 3) Solve for x by isolating the variable, 4) Express your answer as the domain of negativity.

Why is the x-intercept important for function positivity and negativity?

The x-intercept is where the function changes from positive to negative (or vice versa). It's the boundary point that separates the domains of positivity and negativity, making it crucial for determining these intervals.

What are common mistakes when finding function positivity domains?

Common mistakes include: confusing the inequality direction when solving, forgetting to check which side of the x-intercept is positive or negative, and mixing up the relationship between y-values and their position relative to the x-axis.

How do you verify your answer for linear function positivity?

Test a point from your domain in the original function. If you found x > 0.5 is the positive domain, test x = 1: substitute into y = 4x - 2 to get y = 2, which is positive, confirming your answer.

Can a linear function have both positive and negative domains?

Yes, most non-horizontal linear functions have both positive and negative domains separated by their x-intercept. Only horizontal lines (y = constant) are entirely positive, negative, or zero throughout their domain.

What's the relationship between slope and function positivity patterns?

For functions with positive slope: left of x-intercept is negative, right is positive. For negative slope: left of x-intercept is positive, right is negative. The slope determines which side of the intercept has which sign.

Continue Your Math Journey

Master Positivity and Negativity first, then advance to these related topics that build on your fraction skills

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Positivity and Negativity Linear Function Practice Problems

Understanding Positivity and Negativity

Practice Positivity and Negativity

Examples with solutions for Positivity and Negativity

Frequently Asked Questions

How do you find the domain of positivity of a linear function?

What is the difference between positive and negative domains of a function?

How do you solve linear function negativity problems step by step?

Why is the x-intercept important for function positivity and negativity?

What are common mistakes when finding function positivity domains?

How do you verify your answer for linear function positivity?

Can a linear function have both positive and negative domains?

What's the relationship between slope and function positivity patterns?

More Positivity and Negativity Questions