For what x is 1 x + 3 &gt; 2 x - 3 ?

Question

For what x is             1              x        +        3              &amp;gt;          2              x        -        3            ?

Jayvion Tyler · Accepted Answer

Step 1Note that       x    =    -    3   and       x    =    3   are not solutions since one or the other side of the inequality is undefined for these values of x.Split the remaining possibilities into cases:Case                     x        ∈                  (          -          ∞          ,          -          3          )                ∪                  (          3          ,          ∞          )                          x    +    3   and       x    -    3   are both negative or both positive.In either case             (      x      +      3      )              (      x      -      3      )        &amp;gt;    0   so multiply both sides of the inequality by that to get:      x    -    3    &amp;gt;    2    x    +    6  Subtract       x    +    6   from both sides to get:      -    9    &amp;gt;    x  Combining with the conditions of the case, that gives us the solution:      x    ∈          (      -      ∞      ,      -      9      )      Case                     x        ∈                  (          -          3          ,          3          )                    .      x    +    3    &amp;gt;    0   and       x    -    3    &amp;lt;    0  , so             (      x      +      3      )              (      x      -      3      )        &amp;lt;    0  , so multiply both sides of the inequality by this and reverse the inequality to get:      x    -    3    &amp;lt;    2    x    +    6  Step 2Subtract       x    +    6   from both sides to get:      -    9    &amp;lt;    x  Combining with the conditions of the case, that gives us the solution:      x    ∈          (      -      3      ,      3      )      ConclusionPutting the two cases together, the original inequality holds for all      x    ∈          (      -      ∞      ,      -      9      )        ∪          (      -      3      ,      3      )

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