Inequalities

To solve the inequality , we will address it in two parts. Let's first simplify each portion of the inequality.

1. Simplify :
2. Part 1: Solve
   • Subtract from both sides:
   • Divide by 6:
3. Part 2: Solve
   • Rearrange terms:
   • Simplify:
   • Factorize:

The factorized form changes sign at the roots and . We find the intervals where the product is less than zero:

1. Sign Analysis:
   • Interval : Pick , then the expression gives .
   • Interval : Pick , then the expression gives .
   • Interval : Pick , then the expression gives .
2. Hence, in .

Combine the solutions:

• From part 1:
• From part 2:

Overlapping these results, we find . As must be a positive integer, can be any of the numbers 1, 2, 3, 4, 5, 6, or 7.

Thus, there are 7 solutions: the numbers 1, 2, 3, 4, 5, 6, and 7 satisfy both inequalities.

The number of positive integral solutions of the given inequality is 5.

Step 1: Understanding the Concept:
Divide by : .
Step 2: Detailed Explanation:
Let ,

Since and are decreasing functions,
check : , equality
For : both terms larger, sum
For : sum
So
Step 3: Final Answer:
.