1)) A triangle has two sides of length 17.4 and 14.7 What compound inequality describes the possible lengths for the third side x? 1) Write a compound inequality like 1lt xlt 3 square

## 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: $a + b > c$, $a + c > b$, and $b + c > a$.<br /><br />## Step 2: Applying the Theorem<br />### Let $a = 17.4$ and $b = 14.7$. Let $x$ be the length of the third side, $c$. We have: $17.4 + 14.7 > x$, $17.4 + x > 14.7$, and $14.7 + x > 17.4$.<br /><br />## Step 3: Simplifying the Inequalities<br />### $17.4 + 14.7 > x$ simplifies to $32.1 > x$ or $x < 32.1$. $17.4 + x > 14.7$ simplifies to $x > -2.7$. Since lengths are positive, this becomes $x > 0$. $14.7 + x > 17.4$ simplifies to $x > 2.7$.<br /><br />## Step 4: Combining the Inequalities<br />### Combining $x < 32.1$ and $x > 2.7$, we get $2.7 < x < 32.1$.