Question 1193296
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The behavior of the absolute value function changes at the two points where the absolute value expressions are equal to 0.<br>
3x-4=0 --> x=4/3
2x+1=0 --> x=-1/2<br>
Those two values of x divide the domain into three intervals: x less than -1/2, x between -1/2 and 4/3, and x greater than 4/3.  Look for solutions in each of those intervals.<br>
(1) x less than -1/2....<br>
On this interval, {{{abs(3x-4)=-3x+4}}} and {{{abs(2x+1)=-2x-1}}}<br>
{{{2(-3x+4)-3(-2x-1)<7}}}
{{{-6x+8+6x+3<7}}}
{{{11<7}}}<br>
That inequality is clearly never true; so there are no solutions in this interval.<br>
(2) x between -1/2 and 4/3....<br>
On this interval, {{{abs(3x-4)=-3x+4}}} and {{{abs(2x+1)=2x+1}}}<br>
{{{2(-3x+4)-3(2x+1)<7}}}
{{{-6x+8-6x-3<7}}}
{{{-12x+5<7}}}
{{{-2<12x}}}
{{{x>-1/6}}}<br>
x=-1/6 is in this interval, so part of the solution set is x between -1/6 and x=4/3<br>
(3) x greater than 4/3....<br>
On this interval, {{{abs(3x-4)=3x-4}}} and {{{abs(2x+1)=2x+1}}}<br>
{{{2(3x-4)-3(2x+1)<7}}}
{{{6x-8-6x-3<7}}}
{{{-11<7}}}<br>
That inequality is always true, so the entire interval x greater than 4/3 is part of the solution set.<br>
ANSWER: The solution set is (-1/6,infinity)<br>
A graph confirms that answer:<br>
{{{graph(400,400,-3,3,-15,15,2*abs(3x-4)-3*abs(2x+1),7)}}}<br>
The value of the absolute value function (red) is less than 7 (green) everywhere to the right of x=-1/6.<br>