Question 127367
{{{5(x+1)<3x+4}}}


Solve inequalities the same way you solve equations, with two important exceptions.  First, if you multiply or divide both sides of the inequality by a negative number, you must reverse the sense of the inequality.


For example, we know it is true that {{{-2<1}}}, but if we multiply both sides of the inequality by -1, we get 2 on the left and -1 on the right.  The 'less than' symbol is no longer correct, it must be changed to 'greater than' because {{{2>-1}}}


The second exception is that the solution set will be represented by an interval rather than a single value.


{{{5(x+1)<3x+4}}}


Distribute and remove parentheses
{{{5x+5<3x+4}}}


Add -3x to both sides
{{{5x-3x+5<4}}}
{{{2x+5<4}}}


Add -5 to both sides
{{{2x<4-5}}}
{{{2x<-1}}}


Finally, divide both sides by 2
{{{x<-1/2}}} and the solution set is {x| x is real, {{{x<-1/2}}}}.  You can also use interval notation ({{{-infinity}}},{{{-1/2}}}).   Note the parentheses around the interval.  Had the problem included the possiblity of equality, i.e. had it been {{{5(x+1)<=3x+4}}}, then the interval would have been denoted by 
({{{-infinity}}},{{{-1/2}}}].   

Since we never multiplied or divided by a number that is <0, the sense of the inequality remains as it was in the original inequality.