How do you solve and write the following in interval notation: #2/5x

Question

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

Kennedy Adams · Accepted Answer

#x in(-oo,15)uu[20, oo)#
# x in RR-[15,20)#

We know that, #color(red)((I)"if a < b and c > 0# #" then "# #color(red)(ac < bc and "a/c < b/c # #color(blue)((II)"if a <= b and c < 0# #" then"# #color(blue)( ac >= bc and "a/c >= b/c # Here , #2/5x < 6 and -1/2x <= -10# #(i) 2/5x < 6# Multiplying both side by #color(red)(5/2# #color(red)(5/2)2/5x < color(red)( 5/2)6color(red)( to Apply(I)# #=>x < 15# #=>x in (-oo, 15)# #(ii)-1/2x <= -10# Multiplying both sides by #color(blue)((-2)# #color(blue)((-2))(-1/2x) >=color(blue)((-2))(-10)color(blue)(toApply(II)# #=>x >= 20# #=>x in[20 , oo)# From #(i) and (ii) #we get #x in (-oo,15) or x in [20, oo)# #=>x in(-oo,15)uu[20, oo)# #=> x in RR-[15,20)#

Isaiah Anthony · Answer

undefined To solve and write the inequalities in interval notation:

1. $ \frac{2}{5}x < 6 $:
   - Multiply both sides by $ \frac{5}{2} $ to isolate $ x $.
   - $ x < \frac{6 \cdot 5}{2} $
   - $ x < 15 $

Interval notation: $ (-\infty, 15) $

2. $ -\frac{1}{2}x \leq -10 $:
   - Multiply both sides by $ -2 $ to isolate $ x $, remember to flip the inequality sign when multiplying or dividing by a negative number.
   - $ x \geq -10 \cdot (-2) $
   - $ x \geq 20 $

Interval notation: $ [20, +\infty) $

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

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