<u>Given</u>:
The given expression is ![2\left(\frac{3}{4} x+7\right)-3\left(\frac{1}{2} x-5\right)](https://tex.z-dn.net/?f=2%5Cleft%28%5Cfrac%7B3%7D%7B4%7D%20x%2B7%5Cright%29-3%5Cleft%28%5Cfrac%7B1%7D%7B2%7D%20x-5%5Cright%29)
We need to determine the equivalent expression.
<u>Option A:</u> -1
Solving the expression, we get;
![\frac{3}{2} x+14-\frac{3}{2} x+15](https://tex.z-dn.net/?f=%5Cfrac%7B3%7D%7B2%7D%20x%2B14-%5Cfrac%7B3%7D%7B2%7D%20x%2B15)
Simplifying, we get;
![14+15=29](https://tex.z-dn.net/?f=14%2B15%3D29)
Thus, the expression
is not equivalent to -1.
Hence, Option A is not the correct answer.
<u>Option B</u>: 29
Solving the expression, we get;
![\frac{3}{2} x+14-\frac{3}{2} x+15](https://tex.z-dn.net/?f=%5Cfrac%7B3%7D%7B2%7D%20x%2B14-%5Cfrac%7B3%7D%7B2%7D%20x%2B15)
Simplifying, we get;
![14+15=29](https://tex.z-dn.net/?f=14%2B15%3D29)
Thus, the expression
is equivalent to 29.
Hence, Option B is the correct answer.
<u>Option C</u>: ![2\left(\frac{3}{4} x+7\right)+(-3)\left[\frac{1}{2} x+(-5)\right]](https://tex.z-dn.net/?f=2%5Cleft%28%5Cfrac%7B3%7D%7B4%7D%20x%2B7%5Cright%29%2B%28-3%29%5Cleft%5B%5Cfrac%7B1%7D%7B2%7D%20x%2B%28-5%29%5Cright%5D)
Let us rewrite the given expression.
![2\left(\frac{3}{4} x+7\right)+(-3)\left[\frac{1}{2} x+(-5)\right]](https://tex.z-dn.net/?f=2%5Cleft%28%5Cfrac%7B3%7D%7B4%7D%20x%2B7%5Cright%29%2B%28-3%29%5Cleft%5B%5Cfrac%7B1%7D%7B2%7D%20x%2B%28-5%29%5Cright%5D)
Thus, the expression
is equivalent to ![2\left(\frac{3}{4} x+7\right)+(-3)\left[\frac{1}{2} x+(-5)\right]](https://tex.z-dn.net/?f=2%5Cleft%28%5Cfrac%7B3%7D%7B4%7D%20x%2B7%5Cright%29%2B%28-3%29%5Cleft%5B%5Cfrac%7B1%7D%7B2%7D%20x%2B%28-5%29%5Cright%5D)
Thus, Option C is the correct answer.
<u>Option D</u>: ![2\left(\frac{3}{4} x\right)+2(7)+3\left(\frac{1}{2} x\right)+3(-5)](https://tex.z-dn.net/?f=2%5Cleft%28%5Cfrac%7B3%7D%7B4%7D%20x%5Cright%29%2B2%287%29%2B3%5Cleft%28%5Cfrac%7B1%7D%7B2%7D%20x%5Cright%29%2B3%28-5%29)
Rewriting the expression, we get;
![2\left(\frac{3}{4} x+7\right)+(-3)\left[\frac{1}{2} x+(-5)\right]](https://tex.z-dn.net/?f=2%5Cleft%28%5Cfrac%7B3%7D%7B4%7D%20x%2B7%5Cright%29%2B%28-3%29%5Cleft%5B%5Cfrac%7B1%7D%7B2%7D%20x%2B%28-5%29%5Cright%5D)
Hence, the expression
is equivalent not to ![2\left(\frac{3}{4} x\right)+2(7)+3\left(\frac{1}{2} x\right)+3(-5)](https://tex.z-dn.net/?f=2%5Cleft%28%5Cfrac%7B3%7D%7B4%7D%20x%5Cright%29%2B2%287%29%2B3%5Cleft%28%5Cfrac%7B1%7D%7B2%7D%20x%5Cright%29%2B3%28-5%29)
Thus, Option D is not the correct answer.
<u>Option E:</u> ![2\left(\frac{3}{4} x\right)+2(7)+(-3)\left(\frac{1}{2} x\right)+(-3)(-5)](https://tex.z-dn.net/?f=2%5Cleft%28%5Cfrac%7B3%7D%7B4%7D%20x%5Cright%29%2B2%287%29%2B%28-3%29%5Cleft%28%5Cfrac%7B1%7D%7B2%7D%20x%5Cright%29%2B%28-3%29%28-5%29)
Multiplying the terms within the bracket, we get;
![2\left(\frac{3}{4} x\right)+2(7)+(-3)\left(\frac{1}{2} x\right)+(-3)(-5)](https://tex.z-dn.net/?f=2%5Cleft%28%5Cfrac%7B3%7D%7B4%7D%20x%5Cright%29%2B2%287%29%2B%28-3%29%5Cleft%28%5Cfrac%7B1%7D%7B2%7D%20x%5Cright%29%2B%28-3%29%28-5%29)
Hence, the expression
is equivalent to ![2\left(\frac{3}{4} x\right)+2(7)+(-3)\left(\frac{1}{2} x\right)+(-3)(-5)](https://tex.z-dn.net/?f=2%5Cleft%28%5Cfrac%7B3%7D%7B4%7D%20x%5Cright%29%2B2%287%29%2B%28-3%29%5Cleft%28%5Cfrac%7B1%7D%7B2%7D%20x%5Cright%29%2B%28-3%29%28-5%29)
Thus, Option E is the correct answer.