4)) A triangle has two sides of length 6.7 and 15.2. What compound inequality describes the possible lengths for the third side, x? 4) Write a compound inequality like 1lt xlt 3

## Step 1: Triangle Inequality Theorem<br />### The sum of the lengths of any two sides of a triangle must be greater than the length of the third side. This gives us three inequalities: $6.7 + 15.2 > x$, $6.7 + x > 15.2$, and $15.2 + x > 6.7$.<br /><br />## Step 2: Solving the Inequalities<br />### $6.7 + 15.2 > x$ simplifies to $21.9 > x$ or $x < 21.9$. $6.7 + x > 15.2$ simplifies to $x > 15.2 - 6.7$ or $x > 8.5$. $15.2 + x > 6.7$ simplifies to $x > 6.7 - 15.2$ or $x > -8.5$. Since side lengths must be positive, $x > -8.5$ is redundant.<br /><br />## Step 3: Combining the Inequalities<br />### Combining $x < 21.9$ and $x > 8.5$, we get the compound inequality $8.5 < x < 21.9$.