How do you solve x - 2 x + 3 &lt; x + 1 x ?

Question

How do you solve                     x        -        2                    x        +        3              &amp;lt;                  x        +        1            x       ?

Daniel Valdez · Accepted Answer

Step 1The inequality given to you looks like this                    x        -        2                    x        +        3              &amp;lt;                  x        +        1            x      Right from the start, you know that any solution interval must not contain the values of x that will make the two denominators equal to zero.More specifically, you need to have      x    +    3    ≠    0    ⇒    x    ≠    -    3                              and                                  x    ≠    0  With that in mind, use the common denominator of the two fractions, which is equal to             (      x      +      3      )        ⋅    x   , to get rid of the denominators.More specifically, multiply the first fraction by       1    =          x      x       and the second fraction by      1    =                  x        +        3                    x        +        3            This will get you                    x        -        2                    x        +        3              ⋅          x      x        &amp;lt;                  x        +        1            x        ⋅                  x        +        3                    x        +        3                                x                  (          x          -          2          )                            x                  (          x          +          3          )                      &amp;lt;                            (          x          +          1          )                          (          x          +          3          )                            x                  (          x          +          3          )                    This is equivalent to      x          (      x      -      2      )        &amp;lt;          (      x      +      1      )              (      x      +      3      )      Expand the parantheses to get                                                                                          x                  2                                                                          -    2    x    &amp;lt;                                                                                        x                  2                                                                          +    x    +    3    x    +    3  Rearrange the inequality to isolate x on one side      -    6    x    &amp;lt;    3    ⇒    x    &amp;gt;          3                        (          -          6          )                      ⇔    x    &amp;gt;    -          1      2      This means that any value of x that is greater than       -          1      2       , except       x    =    0   , will be a solution to the original inequality.Therefore, the solution interval will be       x    ∈          (      -              1        2            ,      +      ∞      )                            {      0      }

How do you solve <mstyle displaystyle="true"> x - 2

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