47 cm and 39 cm
Explanation
the theorem states that the sum of any two sides of a triangle is greater than or equal to the third side; in symbols, a + b > c
so
Step 1
Let
[tex]\begin{gathered} \text{side}1=45\text{ cm} \\ \text{side}2=9\text{ cm} \\ \text{side}3=\text{ x} \end{gathered}[/tex]
hence
a)
[tex]\begin{gathered} \text{side}1+\text{side2}>x \\ replace \\ 45+9>x \\ 54>x \\ so \\ \text{side}3<54\rightarrow\lbrace35,47,39\rbrace \end{gathered}[/tex]
Step 2
b)
[tex]\begin{gathered} \text{side}1+\text{x}>\text{side}2 \\ replace \\ 45+x>9 \\ \text{subtract 45 in both sides} \\ so \\ 45+x-45>9-45 \\ x>-36\rightarrow all\text{ the options} \end{gathered}[/tex]
Step 3
[tex]\begin{gathered} \text{x+side2}>\text{side}1 \\ replace \\ x+9>45 \\ \text{subtract 9 in both sides} \\ so \\ x+9-9>45-9 \\ x>36\rightarrow\lbrace39,47,54,58\rbrace \end{gathered}[/tex]
hence
the solutions must fit:
[tex]\begin{gathered} \text{side}3<54\rightarrow\lbrace35,47,39\rbrace \\ x>-36\rightarrow all\text{ the options}\lbrace35,58,47,54,39\rbrace \\ x>36 \\ 36therefore, the solution is the intersection of those sets[tex]\lbrace47cm,39cm\rbrace[/tex]
47 cm and 39 cm
I hope this helps you
