Alright, let's solve the given inequality step-by-step.
We start with the inequality:
[tex]\[
\sqrt{\frac{(9 x)(8)}{72}} > 5 \left[(32 + 5) - (30 + 2)\right]
\][/tex]
Step 1: Simplifying inside the square root
First, simplify the expression inside the square root on the left side:
[tex]\[
\frac{(9 x)(8)}{72}
\][/tex]
Notice that [tex]\(72\)[/tex] can be factored as [tex]\(72 = 9 \times 8\)[/tex]. Therefore,
[tex]\[
\frac{(9 x)(8)}{72} = \frac{(9 x)(8)}{9 \times 8} = \frac{9 \times 8 \times x}{9 \times 8} = x
\][/tex]
So the expression inside the square root simplifies to [tex]\(x\)[/tex]. Therefore, the left side becomes:
[tex]\[
\sqrt{x}
\][/tex]
Step 2: Simplifying the right side
Now, simplify the expression on the right side:
[tex]\[
5 \left[(32 + 5) - (30 + 2)\right]
\][/tex]
First, perform the operations inside the parentheses:
[tex]\[
32 + 5 = 37 \quad \text{and} \quad 30 + 2 = 32
\][/tex]
Now, subtract the results:
[tex]\[
37 - 32 = 5
\][/tex]
So the right side simplifies to:
[tex]\[
5 \times 5 = 25
\][/tex]
Step 3: Forming the simplified inequality
With the simplified expressions, rewrite the inequality:
[tex]\[
\sqrt{x} > 25
\][/tex]
Step 4: Solving the inequality
To solve for [tex]\(x\)[/tex], we need to eliminate the square root by squaring both sides of the inequality:
[tex]\[
(\sqrt{x})^2 > 25^2
\][/tex]
This simplifies to:
[tex]\[
x > 625
\][/tex]
So, the solution to the inequality is:
[tex]\[
x > 625
\][/tex]
This is the final answer.
