Chapter 6, Problem 7QC

(a)

To determine

Write and solve three inequalities to express the relationship among the measures of the sides of the triangle shown.

Answer to Problem 7QC

Three possible inequalities are: $x>-5;x>4;x<13$

Explanation of Solution

Given:

The triangle:

  

Concept Used:

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.

The three possible inequalities are: $x+9>4;x+4>9;4+9>x$

Calculation:

Solve all the inequalities separately:

$\begin{array}{l}x+9>4\\ x+9-9>4-9\\ x>-5\end{array}$

$\begin{array}{l}x+4>9\\ x+4-4>9-5\\ x>4\end{array}$

$\begin{array}{l}9+4>x\\ 13>x\\ x<13\end{array}$

Thus, three possible inequalities are: $x>-5;x>4;x<13$

(b)

To determine

Find four possible lengths for the third side of the triangle.

Answer to Problem 7QC

The possible 4 values of the third side are: 6, 9, 10, 11

Explanation of Solution

Given:

The triangle:

  

Concept Used:

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.

Three possible inequalities are: $x>-5;x>4;x<13$

The possible four values of the third side are: 6, 9, 10, 11

Thus, the possible 4 values of the third side are: 6, 9, 10, 11

(c)

To determine

Write a compound inequality for the possible values of x.

Answer to Problem 7QC

The compound inequality for the possible values of x is $5<x<13$

Explanation of Solution

Given:

The triangle:

  

Concept Used:

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.

The compound inequality for the possible values of x is $5<x<13$

Thus, the compound inequality for the possible values of x is $5<x<13$