Question 1018478

How would I solve (3x+2/x+1) > 4 ? Thank you for your help
<pre>{{{(3x + 2)/(x + 1) > 4}}}, with {{{x <> - 1}}} 
3x + 2 > 4(x + 1) ------- Multiplying by LCD, x + 1
3x + 2 > 4x + 4
3x - 4x > 4 - 2
- x > 2
{{{x < 2/(- 1)}}} ------> {{{x < - 2}}}

We now have 2 critical points: - 2 and - 1, and 3 INTERVALS to check:
1)   x < - 2, or x = - 3 (- 3 was chosen since - 3 < - 2)
2)   - 2 < x < - 1, or x = - 1.5 (this is a value between - 2 and - 1)
3)   x > - 1, or x = 0 (0 was chosen since 0 > - 1)

<b><u>1)</b></u>   x < - 2, or x = - 3
     {{{(3 * - 3 + 2)/(- 3 + 1) > 4}}} ----- Substituting - 3 in original inequality
     {{{(- 7)/(- 2) > 4}}}
     {{{7/2 > 4}}} ------- False, so x < - 2 is NOT a solution, as this interval DOES NOT satisfy the inequality

<b><u>2)</b></u>   - 2 < x < - 1, or x = - 1.5
     {{{(3 * - 1.5 + 2)/(- 1.5 + 1) > 4}}} ----- Substituting - 1.5 in original inequality
     {{{(- 2.5)/(- .5) > 4}}}
     {{{5 > 4}}} ------- TRUE, so {{{highlight(highlight_green(highlight(- 2 < x < - 1)))}}} IS a solution, as this interval DOES satisfy the inequality
 
<b><u>3)</b></u>   x > - 1, or x = 0
     {{{(3 * 0 + 2)/(0 + 1) > 4}}} ----- Substituting 0 in original inequality
     {{{2/1 > 4}}}
     {{{2 > 4}}} ------- False, so x > - 1 is NOT a solution, as this interval DOES NOT satisfy the inequality