SpectraSynth

To solve the inequality \(x(x + 6) > 16\), follow these steps:

1. Expand the left side of the inequality:
   \(x^2 + 6x > 16\)

2. Move all terms to one side of the inequality:
   \(x^2 + 6x - 16 > 0\)

3. Factor the quadratic expression:
   \((x + 8)(x - 2) < 0\)

Now we have the factors of the quadratic expression. To determine the intervals where the inequality is satisfied, you can use a sign chart or test points within the intervals. The inequality is satisfied when the expression is greater than 0. Let's analyze the sign of the expression within different intervals:

- When \(x < -8\):
  Both factors \((x + 8)\) and \((x - 2)\) are negative. Their product is positive. So, the inequality holds in this interval.

- When \(-8 < x < 2\):
  \((x + 8)\) is positive, and \((x - 2)\) is negative. Their product is negative. So, the inequality holds in this interval.

- When \(x > 2\):
  Both factors \((x + 8)\) and \((x - 2)\) are positive. Their product is negative. So, the inequality does not hold in this interval.

Now, we can write the solution in interval notation:

Solution: \((- \infty, -8) \cup (-8,2)\)

This indicates that the inequality \(x(x + 6) > 16\) is satisfied when \(x\) is in the intervals from negative infinity to -8 (excluding -8), and from -8 (excluding -8) to 2 (excluding 2).

To graph the function \(f(x) = -3x^3 - 6x^2 + 3x + 6\) and identify its x-intercepts, we can follow these steps:

1. Find the x-intercepts by setting \(f(x)\) to 0 and solving for \(x\):

   \[f(x) = -3x^3 - 6x^2 + 3x + 6 = 0\]

2. Once we find the x-intercepts, we can plot them on the graph.

Let's proceed with these steps:

Step 1: Finding x-intercepts

To find the x-intercepts, we'll solve the equation \(-3x^3 - 6x^2 + 3x + 6 = 0\). This equation can be factored to some extent:

\(-3x^3 - 6x^2 + 3x + 6 = -3(x^3 + 2x^2 - x - 2)\)

Now, let's use a tool like a graphing calculator or a computer algebra system to find the roots of the polynomial \(x^3 + 2x^2 - x - 2\). The roots are approximately -1, 1, and 2.

So, the x-intercepts are: \(x = -1\), \(x = 1\), and \(x = 2\).

Step 2: Graphing the function

Now that we have the x-intercepts, let's proceed to graph the function. Here's a rough sketch of the graph:

```
   ^
   |                 *
   |               *
   |            *  
   |          *      
   |       *         
   |      *           * *
   |    *          *
   | *            *
   |*___________*______________>
            -2    -1   1   2
```

In this graph, the x-intercepts are marked with asterisks. The function \(f(x)\) approaches negative infinity as \(x\) moves towards negative infinity and approaches positive infinity as \(x\) moves towards positive infinity. The graph shows a general shape of the cubic polynomial, with its behavior around the x-intercepts.

Please note that this is a rough sketch and not to scale. For a more accurate and detailed graph, you can use graphing software or a graphing calculator.

You're right, there's a clever trick you can use to solve this problem efficiently.

Start by rearranging the given equation: 
\[ab - 6a + 5b = 37 \Rightarrow ab - 6a + 9b - 4b = 37 \Rightarrow (a+9)(b-4) = 37.\]

Since \(37\) is a prime number, it only has two factors: \(1\) and \(37\). Thus, the possible pairs \((a+9, b-4)\) are \((1,37)\) and \((37,1)\).

Case 1: \(a+9 = 1\) and \(b-4 = 37\)
This implies \(a = -8\) and \(b = 41\). However, since both \(a\) and \(b\) need to be positive integers, this case is not possible.

Case 2: \(a+9 = 37\) and \(b-4 = 1\)
This implies \(a = 28\) and \(b = 5\).

Now, calculate \(|a-b|\):
\[|a-b| = |28 - 5| = 23.\]

So, the smallest possible value of \(|a-b|\) is \(23\).

To solve this question, you need to use the concept of mixing solutions with different concentrations to achieve a desired concentration. Here's how to start approaching each part:

(a) The goal here is to find the amount of red potion that must be added to a certain amount of blue potion to achieve a specific concentration. Let's denote the amount of red potion to be added as \(x\) mL. The concentration of the red potion is 1 magical syrup per volume, and the concentration of the blue potion is \(\frac{1}{2}\) magical syrup per volume. The final concentration we want is 1 magical syrup per volume.

Set up the equation based on the concentration formula:

\[\text{Concentration} \times \text{Volume} = \text{Concentration} \times \text{Volume}.\]

For the red potion:

\[1 \cdot x = \text{magical syrup in red potion}.\]

For the blue potion:

\[\frac{1}{2} \cdot 100 = \text{magical syrup in blue potion}.\]

Since the total volume after mixing is \(100 + x\) mL, and we want the concentration to be 1 magical syrup per volume, the equation becomes:

\[1 \cdot (100 + x) = (\text{magical syrup in red potion} + \text{magical syrup in blue potion}).\]

Solve for \(x\) using this equation.

(b) Similar to part (a), you need to find the amounts of red potion and blue potion to be combined to achieve a certain concentration. Let \(x\) be the amount of red potion (in mL) and \(y\) be the amount of blue potion (in mL). The concentration of the red potion is 1 magical syrup per volume, the concentration of the blue potion is \(\frac{1}{2}\) magical syrup per volume, and the final concentration is 1 magical syrup per volume.

Set up an equation for the amount of magical syrup:

\[1 \cdot x + \frac{1}{2} \cdot y = 1 \cdot (x + y).\]

Solve for \(y\) in terms of \(x\) using this equation, and then substitute this into the total volume equation to solve for \(x\).

(c) Think about the concentrations involved. If you have a red potion with a concentration of 1 magical syrup per volume and a blue potion with a concentration of \(\frac{1}{2}\) magical syrup per volume, is there any way to combine them to achieve a concentration of 1 magical syrup per volume?

Remember that concentrations represent the amount of magical syrup in a certain volume, and you're trying to mix these concentrations to achieve a desired result.