Solve the inequality. Express your solution using interval notation and please show all of your work.

x(x + 6)(8 – x) >= 0

I need help, I guess I'm not understanding the interval notation portion.

I know, that when

0 <= x <= 8
and
x <= -6

that the statement is true.

but when I got my paper back it said that I got it wrong. I'm guessing that because it is not solve using interval notation, but I'm clueless.

the solution would be

x ≤ -6 OR 0 ≤ x ≤ 8

It might have been marked wrong since you used AND

You should have been taught whichever "interval notation" your course is using.

There are different ways to represent an interval. The following is a summary.

The most common one is
x ≤ -6 OR 0 ≤ x ≤ 8
as Reiny has put it.

Others use a specialized notation as follows:
The lower and then upper limits are written in order from left to right, separated by a comma, such as
0,8
The inclusion or exclusion of each limit is indicated by a square bracket or parenthesis accordingly.
In the above example, since 0 and 8 are included in the interval, the interval would be written as:
[0,8]
If for some reason the lower limit is excluded, it would be written as:
(0,8].
A variation of the notation would write the square bracket pointing outwards to mean exclusion, such as:
]0,8] to say the same thing as (0,8].

For the other limit, which goes from -∞ (excluded) to -6 (included), would be written as:
(-∞-6].

To join the two using logical operators, we would use the ∪ for "or", and ∩ for "and".

Thus the final answer for the above problem would be written as:
(-∞,-6]∪[0,8]

(alternatively, ]-∞,-6]∪[0,8] )

To solve the inequality \(x(x + 6)(8 - x) \geq 0\), we can use a sign chart. Here's how you can proceed:

1. Identify the critical points: These are the values where the expression equals zero. Set each factor to zero and solve for \(x\):
\[x = 0, \quad x + 6 = 0 \Rightarrow x = -6, \quad 8 - x = 0 \Rightarrow x = 8.\]

2. Create a sign chart: Draw a number line and mark the critical points on it.
\[
\begin{array}{ c | c c c c }
& x < -6 & -6 < x < 0 & 0 < x < 8 & x > 8 \\
\hline
x(x+6)(8-x) & + & - & + & - \\
\end{array}
\]

3. Test a value in each interval: Choose a value in each interval and evaluate the expression \(x(x + 6)(8 - x)\) to determine the sign. Let's choose -7, -2, 1, and 9 as our test values.

\[
\begin{align*}
x = -7: \quad &(-7)((-7) + 6)(8 - (-7)) = -7 \cdot (-1) \cdot 15 = 105 \quad \text{(positive)} \\
x = -2: \quad &(-2)((-2) + 6)(8 - (-2)) = -2 \cdot 4 \cdot 10 = -80 \quad \text{(negative)} \\
x = 1: \quad &(1)((1) + 6)(8 - 1) = 1 \cdot 7 \cdot 7 = 49 \quad \text{(positive)} \\
x = 9: \quad &(9)((9) + 6)(8 - 9) = 9 \cdot 15 \cdot (-1) = -135 \quad \text{(negative)} \\
\end{align*}
\]

4. Analyze the sign chart: Based on the sign chart and the test values, we determine the intervals where the inequality holds true.

- For \(x < -6\): The expression is positive because there is an odd number of negative factors. So, this interval satisfies the inequality.
- For \(-6 < x < 0\): The expression is negative because there is an even number of negative factors. So, this interval does not satisfy the inequality.
- For \(0 < x < 8\): The expression is positive because there is an odd number of negative factors. So, this interval satisfies the inequality.
- For \(x > 8\): The expression is negative because there is an even number of negative factors. So, this interval does not satisfy the inequality.

5. Write the solution in interval notation: From the analysis above, we can write the solution as the union of the intervals where the inequality holds true: \((- \infty, -6) \cup (0, 8]\).

Therefore, the solution to the inequality \(x(x + 6)(8 - x) \geq 0\) in interval notation is \((- \infty, -6) \cup (0, 8]\).