# How do you solve and write the following in interval notation: 2/5x<6 and -1/2x <= -10?

Jul 15, 2018

$x \in \left(- \infty , 15\right) \cup \left[20 , \infty\right)$

$x \in \mathbb{R} - \left[15 , 20\right)$

#### Explanation:

We know that,

color(red)((I)"if a < b and c > 0 $\text{ then }$ color(red)(ac < bc and "a/c < b/c

color(blue)((II)"if a <= b and c < 0 $\text{ then}$ color(blue)( ac >= bc and "a/c >= b/c

Here ,

$\frac{2}{5} x < 6 \mathmr{and} - \frac{1}{2} x \le - 10$

$\left(i\right) \frac{2}{5} x < 6$
Multiplying both side by color(red)(5/2

color(red)(5/2)2/5x < color(red)( 5/2)6color(red)( to Apply(I)

$\implies x < 15$

$\implies x \in \left(- \infty , 15\right)$

$\left(i i\right) - \frac{1}{2} x \le - 10$

Multiplying both sides by color(blue)((-2)

color(blue)((-2))(-1/2x) >=color(blue)((-2))(-10)color(blue)(toApply(II)

$\implies x \ge 20$

$\implies x \in \left[20 , \infty\right)$

From $\left(i\right) \mathmr{and} \left(i i\right)$we get

$x \in \left(- \infty , 15\right) \mathmr{and} x \in \left[20 , \infty\right)$

$\implies x \in \left(- \infty , 15\right) \cup \left[20 , \infty\right)$

$\implies x \in \mathbb{R} - \left[15 , 20\right)$