How do you solve and write the following in interval notation: 5[5m-(m+8)]> -8(m-6)?

Apr 23, 2017

See the entire solution process below:

Explanation:

First, expand and then group and combine like terms within the brackets [ ]:

$5 \left[5 m - \left(m + 8\right)\right] > - 8 \left(m - 6\right)$

$5 \left[5 m - m - 8\right] > - 8 \left(m - 6\right)$

$5 \left[5 m - 1 m - 8\right] > - 8 \left(m - 6\right)$

$5 \left[\left(5 - 1\right) m - 8\right] > - 8 \left(m - 6\right)$

$5 \left[4 m - 8\right] > - 8 \left(m - 6\right)$

Next, expand the terms within the brackets and parenthesis by multiplying each term within the brackets/parenthesis by the term outside the brackets/parenthesis:

$\textcolor{red}{5} \left[4 m - 8\right] > \textcolor{b l u e}{- 8} \left(m - 6\right)$

$\left(\textcolor{red}{5} \cdot 4 m\right) - \left(\textcolor{red}{5} \cdot 8\right) > \left(\textcolor{b l u e}{- 8} \cdot m\right) + \left(\textcolor{b l u e}{- 8} \cdot - 6\right)$

$20 m - 40 > - 8 m + 48$

Then add $\textcolor{red}{40}$ and $\textcolor{b l u e}{8 m}$ to each side of the inequality to isolate the $m$ term while keeping the inequality balanced:

$20 m - 40 + \textcolor{red}{40} + \textcolor{b l u e}{8 m} > - 8 m + 48 + \textcolor{red}{40} + \textcolor{b l u e}{8 m}$

$20 m + \textcolor{b l u e}{8 m} - 40 + \textcolor{red}{40} > - 8 m + \textcolor{b l u e}{8 m} + 48 + \textcolor{red}{40}$

$\left(20 + \textcolor{b l u e}{8}\right) m - 0 > 0 + 88$

$28 m > 88$

Now, divide each side of the inequality by $\textcolor{red}{28}$ to solve for $m$ while keeping the inequality balanced:

$\frac{28 m}{\textcolor{red}{28}} > \frac{88}{\textcolor{red}{28}}$

$\frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{28}}} m}{\cancel{\textcolor{red}{28}}} > \frac{4 \times 22}{\textcolor{red}{4 \times 7}}$

$m > \frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{4}}} \times 22}{\textcolor{red}{\textcolor{b l a c k}{\cancel{\textcolor{red}{4}}} \times 7}}$

$m > \frac{22}{7}$