<PRE><B>Question 201219</B>

{{{p^3-16p&lt;=0}}} Start with the given inequality



{{{p(p-4)(p+4)&lt;=0}}}  Factor the left side



{{{p(p-4)(p+4)=0}}} Set the left side equal to zero



Set each individual factor equal to zero:


{{{p=0}}}, {{{p-4=0}}} or  {{{p+4=0}}} 


Solve for p in each case:


{{{p=0}}}, {{{p=4}}} or  {{{p=-4}}} 



So our critical values are {{{p=0}}}, {{{p=4}}} and  {{{p=-4}}} 


Now set up a number line and plot the critical values on the number line


{{{number_line( 600, -10, 10,0,4,-4)}}}




So let&#39;s pick some test points that are near the critical values and evaluate them.



Let&#39;s pick a test value that is less than {{{-4}}} (notice how it&#39;s to the left of the leftmost endpoint). This will determine if the interval &lt;font size=&quot;8&quot;&gt;(&lt;/font&gt;*[Tex \LARGE \bf{-\infty,-4}]&lt;font size=&quot;8&quot;&gt;]&lt;/font&gt; is part of the solution set.



So let&#39;s pick {{{p=-5}}}



{{{p(p-4)(p+4)&lt;=0}}} Start with the given inequality



{{{-5(-5-4)(-5+4)&lt;= 0}}} Plug in {{{p=-5}}}



{{{-45&lt;= 0}}} Evaluate and simplify the left side



Since the inequality is true, this means that the interval works. So this tells us that this interval is in our solution set.

   So part our solution in interval notation is &lt;font size=&quot;8&quot;&gt;(&lt;/font&gt;*[Tex \LARGE -\infty,-4]&lt;font size=&quot;8&quot;&gt;]&lt;/font&gt;



   



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Let&#39;s pick a test value that is in between {{{-4}}} and {{{0}}}. This will determine if the interval &lt;font size=&quot;8&quot;&gt;[&lt;/font&gt;*[Tex \LARGE \bf{-4,0}]&lt;font size=&quot;8&quot;&gt;]&lt;/font&gt; is part of the solution set.


So let&#39;s pick {{{p=-2}}}


{{{p(p-4)(p+4)&lt;=0}}} Start with the given inequality



{{{-2(-2-4)(-2+4)&lt;= 0}}} Plug in {{{p=0}}}



{{{24&lt;= 0}}} Evaluate and simplify the left side



Since the inequality is false, this means that we can ignore this interval.


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Now let&#39;s pick a test value that is in between {{{0}}} and {{{4}}} to see if the interval &lt;font size=&quot;8&quot;&gt;[&lt;/font&gt;*[Tex \LARGE \bf{0,4}]&lt;font size=&quot;8&quot;&gt;]&lt;/font&gt; is in the solution set 


So let&#39;s pick {{{p=2}}}


{{{p(p-4)(p+4)&lt;=0}}} Start with the given inequality



{{{2(2-4)(2+4)&lt;= 0}}} Plug in {{{p=5}}}



{{{-24&lt;= 0}}} Evaluate and simplify the left side


Since the inequality is true, this means that the interval &lt;font size=&quot;8&quot;&gt;[&lt;/font&gt;*[Tex \LARGE \bf{0,4}]&lt;font size=&quot;8&quot;&gt;]&lt;/font&gt; is in the solution set.






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Now let&#39;s pick a test value that is greater than {{{4}}} to see if the interval &lt;font size=&quot;8&quot;&gt;[&lt;/font&gt;*[Tex \LARGE \bf{4,\infty}]&lt;font size=&quot;8&quot;&gt;)&lt;/font&gt; is in the solution set 


So let&#39;s pick {{{p=5}}}


{{{p(p-4)(p+4)&lt;=0}}} Start with the given inequality



{{{5(5-4)(5+4)&lt;= 0}}} Plug in {{{p=5}}}



{{{45&lt;= 0}}} Evaluate and simplify the left side


Since the inequality is false, this means that we can ignore this interval.


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Summary:


So the solution in interval notation is:



&lt;font size=&quot;8&quot;&gt;(&lt;/font&gt;*[Tex \LARGE -\infty,-4]&lt;font size=&quot;8&quot;&gt;]&lt;/font&gt; *[Tex \LARGE \cup] &lt;font size=&quot;8&quot;&gt;[&lt;/font&gt;*[Tex \LARGE 0,4]&lt;font size=&quot;8&quot;&gt;]&lt;/font&gt;

</PRE>