Solve (x+1)(x+5) > 0: Finding Values Where Function is Positive

Look at the following function:

$y=\left(x+1\right)\left(x+5\right)$

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

$f(x) > 0$

To solve this problem, we'll perform the following steps:

• Step 1: Identify the roots of the function.
• Step 2: Determine intervals around these roots.
• Step 3: Test intervals to find where the function is positive.

Now, let's work through these steps:

Step 1: Identify the roots. Set $y = 0$ to find the roots for the function:

$(x+1)(x+5) = 0$

This gives $x+1=0$ or $x+5=0$, leading to the roots $x=-1$ and $x=-5$.

Step 2: Determine the intervals. The roots divide the number line into three intervals:

$(- \infty, -5)$, $(-5, -1)$, and $(-1, \infty)$.

Step 3: Test the sign of $f(x)$ in each interval by choosing a test point from each region:

• For $x \in (-\infty, -5)$, choose $x = -6$:

Substitute $x = -6$ into $y = (x+1)(x+5)$:

$y = (-6+1)(-6+5) = (-5)(-1) = 5$, which is positive.

• For $x \in (-5, -1)$, choose $x = -3$:

Substitute $x = -3$ into $y = (x+1)(x+5)$:

$y = (-3+1)(-3+5) = (-2)(2) = -4$, which is negative.

• For $x \in (-1, \infty)$, choose $x = 0$:

Substitute $x = 0$ into $y = (x+1)(x+5)$:

$y = (0+1)(0+5) = 1 \times 5 = 5$, which is positive.

Therefore, the function $f(x) = (x+1)(x+5)$ is positive in the intervals $(-\infty, -5)$ and $(-1, \infty)$. Thus, the solution is:

$x > -1$ or $x < -5$.

Upon reviewing the provided answer choices, the choice that corresponds to this solution is:

Choice 2: $x > -1$ or $x < -5$.