Let f ( x ) = 7 + | 2 x - 1 | . How do you find all x for which f ( x ) &lt; 16 ?

Question

Let             f              (        x        )              =    7    +          |      2      x      -      1      |       . How do you find all x for which             f              (        x        )              &amp;lt;    16   ?

Xzavier Shelton · Accepted Answer

Step 1Given:             f              (        x        )              =    7    +          |      2      x      -      1      |       and             f              (        x        )              &amp;lt;    16  We can write the inequality:      7    +          |      2      x      -      1      |        &amp;lt;    16  Subtract 7 from both sides:            |      2      x      -      1      |        &amp;lt;    9  Because of the piecewise definition of the absolute value function,            |      A      |        =          {                                    A            ;            A            ≥            0                                                -            A            ;            A            &amp;lt;            0                              we can separate the inequality into two inequalities:      -          (      2      x      -      1      )        &amp;lt;    9   and       2    x    -    1    &amp;lt;    9  Multiply both sides of the first inequality by -1:      2    x    -    1    &amp;gt;    -    9   and       2    x    -    1    &amp;lt;    9  Add 1 to both sides of both inequalities:      2    x    &amp;gt;    -    8   and       2    x    &amp;lt;    10  Divide both sides of both inequalities by 2:      x    &amp;gt;    -    4   and       x    &amp;lt;    5  This can be written as:      -    4    &amp;lt;    x    &amp;lt;    5  To check, I will verify that the end points equal 16:      7    +          ∣        2          (      -      4      )        -    1    )          |      =      7      +      |        -    9          ∣        =    16        7    +          |      2              (        5        )            -      1      |        =    7    +          |      9      |        =    16

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