How do you solve the inequality #abs(2x-4)=6#?

Question

How do you solve the inequality #abs(2x-4)>=6#?

Eli Assouline · Accepted Answer

undefined To solve the inequality |2x - 4| ≥ 6, first, isolate the absolute value expression by dividing the inequality into two cases:

1. 2x - 4 ≥ 6 
2. 2x - 4 ≤ -6

Then solve each case separately for x:

1. For 2x - 4 ≥ 6:
   2x - 4 ≥ 6
   2x ≥ 10
   x ≥ 5

2. For 2x - 4 ≤ -6:
   2x - 4 ≤ -6
   2x ≤ -2
   x ≤ -1

So, the solution to the inequality |2x - 4| ≥ 6 is x ≤ -1 or x ≥ 5.

Kennedy Adams · Answer

#x in ]-oo;-1] uu [5;oo[#

in this case

#abs(a)>=b# then #a<=-b and a>=b#

then in #abs(2x-4)>=6#

#2x-4<=-6# and #2x-4>=6#

#2x-4+6<=0# and #2x-4-6>=0#

#2x+2<=0# and #2x-10>=0#

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

then #x in]-oo;-1] uu [5;oo[#

Eli Assouline · Answer

x #>=#5

Move like terms and solve.

Avery Alexander · Answer

#x>=5# or #x<=-1# We need to consider two ranges since the absolute value will make the function change sign at some value. We first need to find this value, which we do by finding when the bit inside the absolute value is equal to #0#: #2x-4=0# #2x=4# #x=2# So, we need to look at when #x>2# and when #x<#2 When #x>2# Since the value inside the absolute value function is positive in this range, we can just remove it: #2x-4>=6# #2x-cancel(4+4)>=6+4# #2x>=10# #x>=10/2# #x>=5# When #x<2# In this range, the function in the absolute value will be negative, so we multiply it by #-1#: #-(2x-4)>=6# #-2x+4>=6# #-2x+cancel(4-4)>=6-4# #-2x>=2# Now we want to divide by #-2#, but we need to be careful since dividing by a negative number flips the sign of the inequality: #x<=2/-2# #x<=-1# Combining the two ranges We can just combine the two ranges to get that #x>=5# or #x<=-1#

How do you solve the inequality #abs(2x-4)&gt;=6#?

How do you solve the inequality #abs(2x-4)>=6#?