How do you solve 2x+3>7 or 2x+9>11?

1 Answer
Dec 31, 2016

Answer:

The solution set is #{x | x>1}#.

Explanation:

For each of these inequalities, there will be a set of #x#-values that make them true. For example, it's pretty clear that large values of #x# (like 1,000) work for both, and negative values (like -1,000) will not work for either.

Since we're asked to solve a "this OR that" pair of inequalities, what we'd like to know are all the #x#-values that will work for at least one of them. To do this, we solve both inequalities for #x#, and then overlap the two solution sets.

Inequality 1:

#2x+3>7" "=>" "2x>4"    "#(subtract 3 from both sides)
#color(white)(2x+3>7)" "=>"     "x>2"    "#(divide both sides by 2)

Inequality 2:

#2x+9>11" "=>" "2x>2"    "#(subtract 9 from both sides)
#color(white)(2x+9>11)" "=>"     "x>1"    "#(divide both sides by 2)

So we need to list all the #x#-values that satisfy either #x>2# or #x>1#.

In this case, if an #x#-value is greater than 2, it will automatically be greater than 1. Thus, the solution set for #2x+3>7# is a subset of the one for #2x+9>11#. That means, all we need to do here is list the solution set for #2x+9>11#, and we're done.

The solution set we need is simply "all #x# such that #x# is greater than 1", or (in math terms):

#{x | x>1}#
or
#x in (1, oo)#