Question

1. solve for x. setup: x=

Explanation:

Step1: Apply the Pythagorean theorem

In a right - triangle, if the two legs are \(a\) and \(b\) and the hypotenuse is \(c\), then \(a^{2}+b^{2}=c^{2}\). Here, \(a = 24\), \(b=3(x + 2)\), and \(c\) is not given explicitly, but we can set up the equation based on the right - triangle relationship. So, \((3(x + 2))^{2}+24^{2}=c^{2}\). However, if we assume this is a right - triangle and we want to find \(x\) using the tangent function (since we have an angle and two sides related to it). Let's assume we are using the fact that \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). If we consider the angle opposite to the side of length \(3(x + 2)\) and adjacent to the side of length \(24\), and assume a right - triangle property. We know that for a right - triangle, we can also use the Pythagorean theorem. \((3(x + 2))^{2}+24^{2}=h^{2}\) (where \(h\) is the hypotenuse). First, expand \((3(x + 2))^{2}\):
\[

$$\begin{align*} (3(x + 2))^{2}&=9(x^{2}+4x + 4)\\ &=9x^{2}+36x+36 \end{align*}$$

\]
And \(24^{2}=576\). So the equation becomes \(9x^{2}+36x + 36+576=h^{2}\). But if we assume this is a special right - triangle or use the ratio of sides. Let's assume we use the fact that if we consider the relationship between the sides directly. If we assume the triangle is a right - triangle and we know that we can set up the following based on the side lengths:
\[

$$\begin{align*} (3(x + 2))^{2}+24^{2}&=( \text{hypotenuse})^{2}\\ 9(x^{2}+4x + 4)+576&=( \text{hypotenuse})^{2}\\ 9x^{2}+36x+36 + 576&=( \text{hypotenuse})^{2}\\ 9x^{2}+36x+612&=( \text{hypotenuse})^{2} \end{align*}$$

\]
If we assume this is a right - triangle and we consider the fact that we can also use the property of similar right - triangles or basic right - triangle side relationships. Let's assume we use the fact that if we consider the side \(3(x + 2)\) and \(24\) and assume a \(3 - 4-5\) right - triangle ratio. If we assume \(\frac{3(x + 2)}{24}=\frac{3}{4}\) (assuming a similar - triangle or ratio relationship). Cross - multiply:
\[

$$\begin{align*} 4\times3(x + 2)&=3\times24\\ 12(x + 2)&=72\\ x+2& = 6 \end{align*}$$

\]

Step2: Solve for \(x\)

Subtract 2 from both sides of the equation \(x + 2=6\).
\[x=6 - 2\]

Answer:

\(x = 4\)