Solve (x-4)(-x+6): Finding Values Where Function is Positive

Q: Why do I need to find the roots first?

The roots are where the function equals zero, creating boundary points that divide the number line into intervals. The function can only change from positive to negative (or vice versa) at these roots.

Q: How do I know which test points to choose?

Pick any convenient number from each interval! For $ x , try x = 0. For \( 4 , try x = 5. For \( x > 6 $, try x = 7. Any point in the interval works.

Q: What if I get confused about the signs?

Remember: negative times negative = positive, but negative times positive = negative. Write out each factor's sign clearly: at x = 5, (5-4) = +1 and (-5+6) = +1, so (+1)(+1) = +1.

Q: Do I include the roots in my final answer?

No! The question asks for $ f(x) > 0 $ (strictly greater than zero). At x = 4 and x = 6, the function equals zero, not greater than zero.

Q: Can I expand the function first instead?

You could expand to get $ y = -x^2 + 10x - 24 $, but the factored form is much easier for solving inequalities! Keep it factored to quickly identify the roots.

Quadratic Inequalities with Sign Analysis

Look at the following function:

$y=\left(x-4\right)\left(-x+6\right)$

Determine for which values of $x$ the following is true:

$f(x) > 0$

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Look at the following function:

$y=\left(x-4\right)\left(-x+6\right)$

Determine for which values of $x$ the following is true:

$f(x) > 0$

Step-by-step solution

To solve this problem, we'll first find the roots of the function $y = (x-4)(-x+6)$ to determine the intervals that we need to examine.

Step 1: Find the roots. Set the function equal to zero: $(x-4)(-x+6) = 0$ .

For $x-4=0$ , we get $x = 4$ .
For $-x+6=0$ , we get $x = 6$ .

Step 2: Identify the intervals created by these roots: $x < 4$ , $4 < x < 6$ , and $x > 6$ .
Step 3: Test a point in each interval to determine the sign of the function.

Select a point from $x < 4$ , say $x = 0$ : $(0-4)(-0+6) = (-4)(6) = -24$ , which is negative.
Select a point from $4 < x < 6$ , say $x = 5$ : $(5-4)(-5+6) = (1)(1) = 1$ , which is positive.
Select a point from $x > 6$ , say $x = 7$ : $(7-4)(-7+6) = (3)(-1) = -3$ , which is negative.

Step 4: Conclude that the function is positive in the interval $4 < x < 6$ .

Therefore, the values of $x$ for which $f(x) > 0$ are those in the interval $\mathbf{4 < x < 6}$ .

Final Answer

$4 < x < 6$

Key Points to Remember

Essential concepts to master this topic

Roots: Set each factor equal to zero to find boundary points
Test Points: Check sign in each interval: (0-4)(6) = -24 is negative
Verification: Function positive only between roots: 4 < x < 6 ✓

Common Mistakes

Avoid these frequent errors

Forgetting to test points in each interval
Don't just find the roots x = 4 and x = 6 and guess the answer = wrong intervals! Without testing points, you can't tell which intervals are positive or negative. Always test a point from each interval to determine the sign.

Practice Quiz

Test your knowledge with interactive questions

The graph of the function below does not intersect the $$ x $$ -axis.

The parabola's vertex is marked A.

Find all values of $$ x $$ where
$f\left(x\right) > 0$ .

FAQ

Everything you need to know about this question

Why do I need to find the roots first?

The roots are where the function equals zero, creating boundary points that divide the number line into intervals. The function can only change from positive to negative (or vice versa) at these roots.

How do I know which test points to choose?

Pick any convenient number from each interval! For $x < 4$ , try x = 0. For $4 < x < 6$ , try x = 5. For $x > 6$ , try x = 7. Any point in the interval works.

What if I get confused about the signs?

Remember: negative times negative = positive, but negative times positive = negative. Write out each factor's sign clearly: at x = 5, (5-4) = +1 and (-5+6) = +1, so (+1)(+1) = +1.

Do I include the roots in my final answer?

No! The question asks for $f(x) > 0$ (strictly greater than zero). At x = 4 and x = 6, the function equals zero, not greater than zero.

Can I expand the function first instead?

You could expand to get $y = -x^2 + 10x - 24$ , but the factored form is much easier for solving inequalities! Keep it factored to quickly identify the roots.

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Solve (x-4)(-x+6): Finding Values Where Function is Positive

Quadratic Inequalities with Sign Analysis

❤️ Continue Your Math Journey!

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why do I need to find the roots first?

How do I know which test points to choose?

What if I get confused about the signs?

Do I include the roots in my final answer?

Can I expand the function first instead?

🌟 Unlock Your Math Potential

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