How do you solve and write the following in interval notation: #|1 + 5x| ≤ 11#?

# | 1+5 x| <= 11 ; 1 + 5 x <= 11 or 5 x <=10 or x <= 2# OR

# | 1+5 x| <= 11 ; 1 + 5 x >= -11 or 5 x >= -12 or x >= -12/5#

or # x >= -2.4 :.# Solution: #-2.4<=x<=2# i.e # x in [-2.4,2]# [Ans]

The equation is

#|1+5x|<=11#

#|1+5x|-11<=0#

When #1+5x=0#

#<=>#, #5x=-1#

#<=>#, #x=-1/5#

Let's build a sign chart

#color(white)(aaaaaa)##x##color(white)(aaaaaaaaa)##-oo##color(white)(aaaaaaaaa)##-1/5##color(white)(aaaaaaaaaa)##+oo#

#color(white)(aaaaaa)##1+5x##color(white)(aaaaaaaaaa)##-##color(white)(aaaaaaa)##0##color(white)(aaaaa)##+#

#color(white)(aaaaaa)##|1+5x|##color(white)(aaaaaaa)##-1-5x##color(white)(aaaaaa)####color(white)(aaa)##1+5x#

#color(white)(aaaaaa)##|1+5x|-11##color(white)(aaa)##-12-5x##color(white)(aaa)####color(white)(aaa)##-10+5x#

Therefore,

#-12-5x<=0#, #<=>#, #5x>=-12#, #<=>#, #x<=-12/5#

#x in (-oo, -1/5)#

and

#-10+5x<=0#, #<=>#, #x<=2#

#x in (-1/5, +oo)#

The solution is

# x in [-12/5, 2]#

graph{|1+5x|-11 [-16.09, 15.95, -12.05, 3.97]}

#" "#
#color(green)("Step 1:"#

Given the inequality: #color(red)(|1+5x|<=11#

There are two possible cases:

Case 1: #color(red)(1+5x<=11# #And#

Case 2: #color(red)(1+5x>=-11#

Case 1:

#1+5x<=11#

Subtract #color(red)1# from both sides of the inequality.

#1+5x-color(red)1<=11-color(red)1#

#cancel 1+5x-color(red)cancel 1<=11-color(red)1#

#5x<= 10#

Divide both sides of the inequality by #color(red)5#

#(5x)/color(red)5<= 10/color(red)5#

#(cancel 5x)/color(red)cancel 5<= cancel 10^color(red)2/color(red)cancel 5#

#:. x <=2 " " # Solution 1

Case 2:

#1+5x>=-11#

Subtract #color(red)1# from both sides of the inequality.

#1+5x-color(red)(1)>=-11-color(red)(1#

#cancel 1+5x-color(red)(cancel 1)>=-11-color(red)(1#

#5x>=-12#

Divide both sides of the inequality by #color(red)5#

#(5x)/color(red)5>=-(12)/color(red)5#

#(cancel 5x)/color(red)cancel 5>=-(12)/color(red)5#

#x>=-12/5" "# Solution 2

Combine both the solutions: Solution 1 and Solution 2

#color(green)( :. x <=2 and x>=-12/5" "# is our final solution.

You can also rewrite the combined solution as:

#color(blue)(-12/5<=x<=2#

In Interval Notation: #color(blue)([-12/5,2]#

#color(green)("Step 2:"#

Graph the inequality to verify our solution:

Hope it helps.