Question 135456
*[Tex \LARGE x^2-16\le0]  Start with the given inequality



*[Tex \LARGE (x+4)(x-4)\le0]  Factor the left side




*[Tex \LARGE (x+4)(x-4)=0]  Set the left side equal to zero



*[Tex \LARGE  x=-4  \textrm{ or }   x=4  ] Solve for x in each case



So this means that the critical values are



*[Tex \LARGE  x=-4  \textrm{ or }   x=4  ]



Now plot the critical values on a number line


{{{number_line( 600, -10, 10, -4,4 )}}}




So let's evaluate a point that is to the left of *[Tex \LARGE  x=-4 ] (which is the left most endpoint). Let's evaluate the value {{{x=-5}}}

  

*[Tex \LARGE (x+4)(x-4)<=0]  Start with the given inequality


*[Tex \LARGE (-5+4)(-5-4)<=0]  Plug in {{{x=-5}}}


*[Tex \LARGE 9<= 0]  Evaluate and simplify



 Since *[Tex \LARGE 9<= 0] is false, this means that the interval does not work. So that means that this interval is not in the solution set and can be ignored.

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Now let's test a point that is in between the critical values *[Tex \LARGE  x=-4 ] and *[Tex \LARGE  x=4 ]



*[Tex \LARGE (x+4)(x-4)<=0]  Start with the given inequality


*[Tex \LARGE (0+4)(0-4)<=0]  Plug in {{{x=0}}}


*[Tex \LARGE -16<= 0]  Evaluate and simplify




Since *[Tex \LARGE -16<= 0] is true, this means that one part of the solution in interval notation is *[Tex \LARGE \left[-4,4\right]]

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So let's evaluate a point that is to the right of *[Tex \LARGE  x=4 ] (which is the right most endpoint). Let's evaluate the value {{{x=5}}}

  

*[Tex \LARGE (x+4)(x-4)<=0]  Start with the given inequality


*[Tex \LARGE (5+4)(5-4)<=0]  Plug in {{{x=5}}}


*[Tex \LARGE 9<= 0]  Evaluate and simplify



 Since *[Tex \LARGE 9<= 0] is false, this means that the interval does not work. So that means that this interval is not in the solution set and can be ignored.

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So the answer in interval notation is *[Tex \LARGE \left[-4,4\right]]