# How do you solve the inequality -a/7+1/7>1/14?

Jan 19, 2017

$\textcolor{g r e e n}{a < \frac{1}{2}}$

#### Explanation:

A fraction consists of $\left(\text{count")/("size indicator") ->("numerator")/("denominator}\right)$

You can not directly compare the counts (numerators) unless the size indicators (denominators) as the same.

$\textcolor{g r e e n}{\left[- \frac{a}{7} \textcolor{red}{\times 1}\right] + \left[\frac{1}{7} \textcolor{red}{\times 1}\right] > \frac{1}{14}}$

$\textcolor{g r e e n}{\left[- \frac{a}{7} \textcolor{red}{\times \frac{2}{2}}\right] + \left[\frac{1}{7} \textcolor{red}{\times \frac{2}{2}}\right] > \frac{1}{14}}$

color(green)(" "-(2a)/14" "+" "2/14" ">1/14

Now that the denominators are all the same the inequality is still true if we compare only the counts (numerators).

$\text{ } \textcolor{g r e e n}{- 2 a + 2 > 1}$

Divide both sides by 2

$\text{ } \textcolor{g r e e n}{- a + 1 > \frac{1}{2}}$

Subtract 1 from both sides

$\text{ } \textcolor{g r e e n}{- a > \frac{1}{2} - 1}$

$\text{ } \textcolor{g r e e n}{- a > - \frac{1}{2}}$

Multiply both sides by (-1) and turn the inequality round the other way. You always turn the inequality if multiply by any negative value.

$\text{ } \textcolor{g r e e n}{a < \frac{1}{2}}$