Question

complete each inequality so that it describes possible lengths to make a triangle.
a. 6, 8, x
type the answers in the boxes.
<x<
b. 13, 17, y
<y<
c. 5,5, z
<z<
d. 98, 100, a
<a<

Explanation:

Step1: Recall the triangle inequality theorem

The sum of the lengths of any two sides of a triangle must be greater than the length of the third side. For sides \(a\), \(b\), and \(c\), \(|a - b|\lt c\lt a + b\)

Step2: Solve for \(x\) (case \(a = 6\), \(b = 8\))

• Lower - bound: \(|6 - 8|=2\)
• Upper - bound: \(6 + 8 = 14\)

So \(2\lt x\lt14\)

Step3: Solve for \(y\) (case \(a = 13\), \(b = 17\))

• Lower - bound: \(|13 - 17|=4\)
• Upper - bound: \(13+17 = 30\)

So \(4\lt y\lt30\)

Step4: Solve for \(z\) (case \(a = 5\), \(b = 5\))

• Lower - bound: \(|5 - 5|=0\)
• Upper - bound: \(5 + 5=10\)

So \(0\lt z\lt10\)

Step5: Solve for \(a\) (case \(a = 98\), \(b = 100\))

• Lower - bound: \(|98 - 100|=2\)
• Upper - bound: \(98 + 100=198\)

So \(2\lt a\lt198\)

Answer:

a. \(2\lt x\lt14\)
b. \(4\lt y\lt30\)
c. \(0\lt z\lt10\)
d. \(2\lt a\lt198\)