Question

if the sides of a triangle are 3,7, and x, which of the following best describes x?
○ $x > 4$
○ $4 \leq x \leq 10$
○ $x < 10$
○ $4 < x < 10$
5 multiple choice 4 points
which of the following sets of numbers could represent the lengths of the sides of a triangle?
○ 3,3,5
○ 5,5,15
○ 4,4,8
○ 2,2,5

Explanation:

First Question

Step1: Apply triangle inequality

For sides $a,b,c$, $a+b>c$, $a+c>b$, $b+c>a$

Step2: Set up inequalities

$3+7>x$, $3+x>7$, $7+x>3$

Step3: Simplify each inequality

$10>x$, $x>4$, $x>-4$

Step4: Combine valid inequalities

$4 < x < 10$

---

Second Question

Step1: Test option A: 3,3,5

$3+3>5$ ($6>5$), $3+5>3$ ($8>3$), $3+5>3$ ($8>3$)

Step2: Test option B:5,5,15

$5+5=10<15$ (fails inequality)

Step3: Test option C:4,4,8

$4+4=8$ (fails strict inequality)

Step4: Test option D:2,2,5

$2+2=4<5$ (fails inequality)

Step5: Identify valid set

Only 3,3,5 satisfies all rules

Answer:

D. $4 < x < 10$