For what x is 3 x + 3 &gt; 4 x x - 2 ?

Question

For what x is             3              x        +        3              &amp;gt;                  4        x                    x        -        2            ?

Daniel Valdez · Accepted Answer

Step 1To simplify the inequality, we are going to perform cross-multiplication of the denominator to avoid dealing with fractions. However, the sign of the inequality might be flipped if we multiply both sides by a negative expression. Hence, we consider x on 3 separate intervals, namely,             (      -      ∞      ,      -      3      )      ,             (      -      3      ,      2      )      ,             (      2      ,      ∞      )      . Note that x cannot -3 or 2. Now, we wish to find the values of x which satisfies                     3                    x        +        3              &amp;gt;                  4        x                    x        -        2            .For the case of       x    ∈          (      -      ∞      ,      -      3      )      , both       x    +    3   and       x    -    2   are negative.      3          (      x      -      2      )        &amp;gt;    4    x          (      x      +      3      )      The sign remains the same as the product of 2 negative expressions is positive.      3    x    -    6    &amp;gt;    4          x      2        +    12    x        0    &amp;gt;    4          x      2        +    9    x    +    6        0    &amp;gt;          x      2        +          9      4        x    +          3      2      Completing the square,       0    &amp;gt;                  (        x        +                  9          8                )            2        +                  15                    64            .Step 2Since the RHS is always       ≥          15      64      , no real value of x satisfies       3          (      x      -      2      )        &amp;gt;    4    x          (      x      +      3      )      .Therefore, no value of       x    ∈          (      -      ∞      ,      -      3      )       satisfies                     3                    x        +        3              &amp;gt;                  4        x                    x        -        2            .For the case of       x    ∈          (      -      3      ,      2      )      ,       x    +    3   is positive while       x    -    2   is negative.      3          (      x      -      2      )        &amp;lt;    4    x          (      x      +      3      )      The sign flips as the product of a positive and a negative expressions is negative.Completing the square,       0    &amp;lt;                  (        x        +                  9          8                )            2        +                  15                    64            Since the RHS is always       ≥          15      64        &amp;gt;    0  , all real value of x satisfies       3          (      x      -      2      )        &amp;gt;    4    x          (      x      +      3      )      .Therefore, all values of       x    ∈          (      -      3      ,      2      )       satisfies                     3                    x        +        3              &amp;gt;                  4        x                    x        -        2            .Step 3For the case of       x    ∈          (      2      ,      ∞      )      , both       x    +    3   and       x    -    2   are negative.      3          (      x      -      2      )        &amp;gt;    4    x          (      x      +      3      )      The sign remains the same as the product of 2 negative expressions is positive.Completing the square,       0    &amp;gt;                  (        x        +                  9          8                )            2        +                  15                    64            .Since the RHS is always       ≥          15      64      , no real value of x satisfies       3          (      x      -      2      )        &amp;gt;    4    x          (      x      +      3      )      .Therefore, no value of       x    ∈          (      2      ,      ∞      )       satisfies                     3                    x        +        3              &amp;gt;                  4        x                    x        -        2            .Combining the solutions on all 3 intervals, we get                     3                    x        +        3              &amp;gt;                  4        x                    x        -        2            .if and only if       x    ∈          (      -      3      ,      2      )

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