Question 616627
help Solve the inequality.
<pre>
{{{(x-5)/(x+4)}}} < 1 

Subtract 1 from both sides:

{{{(x-5)/(x+4)}}} - 1 < 0

{{{(x-5)/(x+4)}}} - {{{1/1}}} < 0

Get an LCD of (x+4) and multiply the {{{1/1}}} by {{{(x+4)/(x+4)}}} < 0

{{{(x-5)/(x+4)}}} - {{{1/1}}}·{{{(x+4)/(x+4)}}} < 0

{{{(x-5)/(x+4)}}} - {{{(x+4)/(x+4)}}} < 0

Combine the numerators over the LCD:

{{{((x-5)-(x+4))/(x+4)}}} < 0

{{{(x-5-x-4)/(x+4)}}} < 0

{{{(-9)/(x+4)}}} < 0

To find the critical number, set x+4 = 0, getting x = -4

Since there is only one critical number we have only two
intervals to test:

Interval          | (-oo,-4)  |  (-4, oo) |
Test value        |    -5     |      0    |
Sign of left side |           |           |

The left side is  {{{(-9)/(x+4)}}}

We substitute test value -5 in {{{(-9)/(x+4)}}}
{{{(-9)/(-5+4)}}} = 9 which is positive so we put a +

Interval          | (-&#8734;,-4)  |  (-4, &#8734;) |
Test value        |    -5    |      0   |
Sign of left side |     +    |          |

We substitute test value 0 in {{{(-9)/(x+4)}}}
{{{(-9)/(0+4)}}} = {{{-9/4}}} which is negative so we put a -

Interval          | (-&#8734;,-4)  |  (-4, &#8734;) |
Test value        |    -5    |      0   |
Sign of left side |     +    |      -   |

Since it's < 0 we choose the interval (-4, &#8734;) as the solution

Edwin</pre>