Answer
See proof: solve the equations given by the two conditions, plug in the greatest and least values of a and b into the final condition.
Work Step by Step
Solve the equations given by the two conditions
The first condition: $|a − 5| < \frac{1}{2}$
$a-5-\frac{1}{2}$
$a<5\frac{1}{2}$ and $a>4\frac{1}{2}$
The second condition: $|b − 8| < \frac{1}{2}$
$b-8-\frac{1}{2}$
$b<8\frac{1}{2}$ and $b>7\frac{1}{2}$
The hint is to use the triangle inequality $(|a + b|≤|a| +|b|)$, but I did not use the hint.
To prove $|(a + b) − 13| < 1$, we take the greatest values that a and b could possibly be.
$|<5\frac{1}{2}+<8\frac{1}{2}-13|$
$|<14-13|$
To prove $|(a+b)−13|<1$, we also take the least values that a and b could possibly be.
$|>4\frac{1}{2}+>7\frac{1}{2}-13|$
$|>12-13|$
These value has to be $<1$.
