Question 1049247
.
|x+2|<2x+7.
~~~~~~~~~~~~~~~~~~~~


|x+2|<2x+7. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(1)


<pre>
1.  If x+2 >= 0, then  |x+2| = x+2.

    Hence, in the domain x >= -2 the inequality (1) takes the form

    x + 2 < 2x + 7,   or  2-7 < x,  which is  x > -5.

    In other words, in the domain x >= -2 the solution set is x > -5.
    Thus for the domain x > -2 the solution set is the intersection [{{{-2}}},{{{infinity}}})  and  ({{{-5}}},{{{infinity}}}), i.e.  [{{{-2}}},{{{infinity}}}).


2.  If x+2 < 0, then  |x+2| = -(x+2).

    Hence, in the domain x < -2 the inequality (1) takes the form

    -(x + 2) < 2x + 7,   or  -x - 2 < 2x +7,  or  -2-7 < 3x,  or  -9 < 3x, which is  x > -3.

    In other words, in the domain x < -2 the solution set is x > -3.
    Thus for the domain x < -2 the solution set is the intersection ({{{-infinity}}},{{{-2}}})  and  ({{{-3}}},{{{infinity}}}), i.e.  ({{{-3}}},{{{-2}}}).


3.  Finally, combining these two cases, we obtain the solution set as  ({{{-3}}},{{{-2}}}) U [{{{-2}}},{{{infinity}}},  i.e.  ({{{-3}}},{{{infinity}}}).
</pre>

<U>Answer</U>.  The solution set to inequality (1) is  ({{{-3}}},{{{infinity}}}).


{{{graph( 330, 330, -10.5, 10.5, -10.5, 10.5,
          abs(x+2), 2x+7
)}}}


Plots y = |x+2| and y = 2x+7