In this image, we have a right triangle with two sides given as 14.2 inches, and the perimeter of the triangle must be less than 51.3 inches. One side of the triangle is denoted as $x$ inches, which we need to find by setting up an inequality and solving it.

Step-by-step process:

1. Perimeter of the triangle: The perimeter $P$ of a triangle is the sum of the lengths of all three sides. Here, the sides are $14.2$, $14.2$, and $x$. Thus, the perimeter can be expressed as: $P = 14.2 + 14.2 + x$ This gives: $P = 28.4 + x$ We are told that the perimeter must be less than 51.3 inches, so: $28.4 + x < 51.3$

2. Solving the inequality: To solve for $x$, subtract 28.4 from both sides: $x < 51.3 - 28.4$ $x < 22.9$

Thus, the inequality is $x < 22.9$, and the solution to this inequality is that $x$ must be less than 22.9 inches.

Final answers:

• Inequality: $28.4 + x < 51.3$
• Solution: $x < 22.9$

Let me know if you'd like further details or have any questions!

Here are five related questions for further exploration:

1. How would the inequality change if the perimeter condition was "less than or equal to 51.3 inches"?
2. What is the range of possible values for $x$ in this case?
3. How would you find the area of this triangle if $x$ was given?
4. Can this triangle still be classified as isosceles based on the information provided?
5. How does the Pythagorean Theorem apply to this triangle if the two equal sides form the legs?

Tip: Always double-check units when solving perimeter or area problems to avoid confusion!