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

Question

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

Sawyer Day · Accepted Answer

Step 1First divide both sides by 3 to get:            1              x        +        3              &amp;gt;          1              x        -        2            We can ignore the values       x    =    -    3   and       x    =    2   since one or the other side of the inequality is undefined for these values. That leaves             (      -      ∞      ,      -      3      )        ∪          (      -      3      ,      2      )        ∪          (      2      ,      ∞      )      .It is probably easiest to split this into cases as follows:Case                     x        ∈                  (          -          ∞          ,          -          3          )                                (      x      +      3      )        &amp;lt;    0   and             (      x      -      2      )        &amp;lt;    0   so             (      x      +      3      )              (      x      -      2      )        &amp;gt;    0  .Multiply both sides of the inequality by             (      x      +      3      )              (      x      -      2      )       to get:      x    -    2    &amp;gt;    x    +    3  Subtract x from both sides to get       -    2    &amp;gt;    3  , which is false.So there is no solution for       x    ∈          (      -      ∞      ,      -      3      )      .Step 2Case                     x        ∈                  (          -          3          ,          2          )                    .            (      x      +      3      )        &amp;gt;    0   and             (      x      -      2      )        &amp;lt;    0   so             (      x      +      3      )              (      x      -      2      )        &amp;lt;    0  .Multiply both sides of the inequality by             (      x      +      3      )              (      x      -      2      )       and reverse the inequality (since we&#039;re multiplying by a negative value) to get:      x    -    2    &amp;lt;    x    +    3  Subtract x from both sides to get       -    2    &amp;lt;    3  , which is true.So the inequality is true for all       x    ∈          (      -      3      ,      2      )      Case                     x        ∈                  (          2          ,          ∞          )                                (      x      +      3      )        &amp;gt;    0   and             (      x      -      2      )        &amp;gt;    0   so             (      x      +      3      )              (      x      -      2      )        &amp;gt;    0  .Multiply both sides of the inequality by             (      x      +      3      )              (      x      -      2      )       to get:      x    -    2    &amp;gt;    x    +    3  Subtract x from both sides to get       -    2    &amp;gt;    3  , which is false.So there is no solution with       x    ∈          (      2      ,      ∞      )      .

For what x is <mstyle displaystyle="true"> 1 x +

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