Q52. - Algebra 2

Question

For exercises 51 and 52, use the following information. 

The Triangle Inequality Theorem states that the sum of the measures of any two sides of a triangle is greater than the measure of the third side.

51. Write a compound inequality to describe the range of possible measures for side $\mathbf{c}$ in terms of $\mathbf{a}$ and $\mathbf{b}$. Assume that $\mathbf{a}>\mathbf{b}>\mathbf{c}$. (Hint: Solve each inequality you wrote in exercise 51 for c)

 

Step-by-Step Solution

Answer

A compound inequality to describe the range of possible measures for side $c$ in terms of $a$and $b$ is$\mathbf{a}\mathbf{-}\mathbf{b}<\mathbf{c}<\mathbf{a}\mathbf{+}\mathbf{b}$.

 

Step 1 &ndash; Description of step.

Solve$a+c>b$. Subtract a to solve for c from each side of $a+c>b$to solve for c.

$\begin{array}{l}a+c-a>b-a\\ c>b-a\end{array}$

Step 2 &ndash; Description of step.

Solve $b+c>a$. Subtract $b$ from each side of $b+c>a$ $b+c>a$ to solve for $c$.

$\begin{array}{l}b+c-b>a-b\\ c>a-b\end{array}$

Step 3 &ndash; Description of step.

From $a+b>c$, $c>b-a$ and $c>a-b$ it can be concluded that $a-b<c<a+b$.

 Therefore, a compound inequality to describe the range of possible measures for side$c$ in terms of $a$ and $b$ is $a-b<c<a+b$.