Question 1209364
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{{{abs(x-4)+2(x+3)<=11+5abs(x+7)+3x+8}}}<br>
Simplify the inequality....<br>
{{{abs(x-4)+2x+6<=5abs(x+7)+3x+19}}}
{{{abs(x-4)-5abs(x+7)<=x+13}}}<br>
The behavior of the function changes when the arguments of the absolute value expressions are equal to 0 -- at x=4 and x=-7.  That divides the number line into three intervals: (-infinity,-7], [-7,4], and [4,infinity)<br>
Find the value on each interval that satisfy the inequality.<br>
(1) (-infinity,-7]<br>
On this interval, {{{abs(x-4)=-x+4}}} and {{{abs(x+7)=-x-7}}}<br>
{{{(-x+4)-5(-x-7)<=x+13}}}
{{{-x+4+5x+35<=x+13}}}
{{{4x+39<=x+13}}}
{{{3x<=-26}}}
{{{x<=-26/3}}}<br>
Of the values of x on the given interval (-infinity,-7], the ones that satisfy the inequality are those less than or equal to -26/3.<br>
First part of solution set: (-infinity,-26/3]<br>
(2) [-7,4]<br>
On this interval, {{{abs(x-4)=-x+4}}} and {{{abs(x+7)=x+7}}}<br>
{{{(-x+4)-5(x+7)<=x+13}}}
{{{-x+4-5x-35<=x+13}}}
{{{-6x-31<=x+13}}}
{{{7x>=-44}}}
{{{x>=-44/7}}}<br>
Of the value of x on the given interval [-7,4], the ones that satisfy the inequality are those greater than or equal to -44/7.<br>
Second part of solution set: [-44/7,4]<br>
(3) [4,infinity)<br>
On this interval, {{{abs(x-4)=x-4}}} and {{{abs(x+7)=x+7}}}<br>
{{{(x-4)-5(x+7)<=x+13}}}
{{{x-4-5x-35<=x+13}}}
{{{-4x-39<=x+13}}}
{{{5x>=-52}}}
{{{x>=-52/5}}}<br>
All of the values of x on the given interval [4,infinity) are greater than or equal to -52/5.<br>
Third part of solution set: [4,infinity)<br>
ANSWER: the complete solution set is (-infinity,-26/3] U [-44/7,infinity)<br>
The simplified form of the inequality I used is {{{abs(x-4)-5abs(x+7)<=x+13}}}, which is equivalent to {{{abs(x-4)-5abs(x+7)-(x+13)<=0}}}.<br>
Here is a graph of that function showing the value is less than or equal to 0 on  (-infinity,-26/3] U [-44/7,infinity).<br>
{{{graph(400,400,-10,10,-10,10,abs(x-4)-5abs(x+7)-(x+13))}}}<br>