Question

the side lengths of a triangle are 10, 19, and x units.
(the figure above gives only one possible configuration for the triangle.)
determine the possible values of x.
write your answer as an inequality.
use x only once in your inequality.

Explanation:

Step1: Recall Triangle Inequality Theorem

The sum of any two sides of a triangle must be greater than the third side. So we have three inequalities: \(10 + 19>x\), \(10 + x>19\), and \(19 + x>10\). But \(19 + x>10\) is always true for positive \(x\), so we focus on the first two.

Step2: Simplify \(10 + x>19\)

Subtract 10 from both sides: \(x>19 - 10\) which simplifies to \(x>9\).

Step3: Simplify \(10 + 19>x\)

Calculate \(10 + 19 = 29\), so \(x<29\).

Step4: Combine the inequalities

From Step2 and Step3, we get \(9

Answer:

\(9 < x < 29\)