## Solution 37 Page 15 Math Workbook 10 – Kite>

Topic

Find the sets $$A = \left[ { – 1;7} \right],B = \left( {m – 1;m + 5} \right)$$ where m is a real parameter. Find m to

a) $$B \subset A$$

b) $$A \cap B = \emptyset$$

Detailed explanation

a)

To $$B \subset A$$ then $$\left\{ {\begin{array}{*{20}{c}}{m – 1 \ge – 1}\\{m + 5 \le 7} \end{array} \Leftrightarrow \left\{ {\begin{array}{*{20}{c}}{m \ge 0}\\{m \le 2}\end{array}} \right.} \right.\Leftrightarrow 0 \le m \le 2$$

So for m satisfying $$0 \le m \le 2$$ then $$B \subset A$$

b)

To $$A \cap B = \emptyset$$ then $$\left[{\begin{array}{*{20}{c}}{m–1\ge7}\\{m+5\le–)1}\end{array}}\right\Leftrightarrow\left[{\begin{array}{*{20}{c}}{m\ge8}\\{m\le–6}\end{array}}\right$$[{\begin{array}{*{20}{c}}{m–1\ge7}\\{m+5\le –1}\end{array}}\right\Leftrightarrow\left[{\begin{array}{*{20}{c}}{m\ge8}\\{m\le –6}\end{array}}\right\)

So for m satisfying $$m \le – 6$$ or $$m \ge 8$$ then $$A \cap B = \emptyset$$