Solve f(x) = x² - 6x + 8 < 0: Graphical Analysis with Linear Function

The following functions are graphed below:

$f(x)=x^2-6x+8$

$g(x)=4x-17$

For which values of x is
$f(x)<0$ true?

To solve the inequality $f(x) = x^2 - 6x + 8 < 0$, we follow these steps:

• Step 1: Find the roots of the equation $f(x) = 0$ using the factoring method.
• Step 2: Determine the intervals formed by these roots.
• Step 3: Test points from each interval to determine where $f(x) < 0$.

Step 1: Factor the quadratic equation $x^2 - 6x + 8 = 0$.
Factoring gives: $(x - 2)(x - 4) = 0$.

Thus, the roots are $x = 2$ and $x = 4$.

Step 2: The roots divide the number line into three intervals: $x < 2$, $2 < x < 4$, and $x > 4$.

Step 3: Choose a test point from each interval and plug it into $f(x)$:

• For $x < 2$, choose $x = 1$: $f(1) = 1^2 - 6(1) + 8 = 1 - 6 + 8 = 3$, which is positive, so $f(x) \geq 0$.
• For $2 < x < 4$, choose $x = 3$: $f(3) = 3^2 - 6(3) + 8 = 9 - 18 + 8 = -1$, which is negative, so $f(x) < 0$.
• For $x > 4$, choose $x = 5$: $f(5) = 5^2 - 6(5) + 8 = 25 - 30 + 8 = 3$, which is positive, so $f(x) \geq 0$.

Therefore, the interval where $f(x) < 0$ is $2 < x < 4$.

The correct choice is:

$2 < x < 4$