How do you solve 1 2 x 2 - x &gt; 4 ?

Question

How do you solve                     1                    2                    x              2              -    x    &amp;gt;    4   ?

tennispopj8 · Accepted Answer

Step 1Bring everything to one side.            1      2              x      2        -    x    -    4    &amp;gt;    0  Factor that side if possible.Factor the             1      2       out of all three terms.            1      2              (              x        2            -      2      x      -      8      )        &amp;gt;    0  Recognize that      –    4    ×    2    =    8    ,   and       –    4    +    2    =    –    2    .              1      2              (      x      -      4      )              (      x      +      2      )        &amp;gt;    0  Divide both sides by             1      2                  (      x      -      4      )              (      x      +      2      )        &amp;gt;    0  We now have a product of two factors on the left:       x    -    4   and       x    +    2   . We are interested in when this product is positive             (      &amp;gt;      0      )        .  Both factors depend on x. Therefore, their product will be positive when x is small/large enough to make the factors either both negative or both positive.The 1st factor,       x    -    4   is negative when       x    &amp;lt;    4  The 2nd factor,       x    +    2   , is negative when       x    &amp;lt;    –    2  So both factors will be negative when       x    &amp;lt;    –    2  Similarly, both factors will be positive when       x    &amp;gt;    4  Our solution is all x in both these regions:       x    &amp;lt;    –    2    ∪    x    &amp;gt;    4    .

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