How to solve | 2 x + 3 | ≥ - 13 ?

Question

How to solve             |              2        x        +        3            |        ≥    -    13  ?

trackrunner92yuy · Accepted Answer

By definition,             |      z      |        ≥    0    ∀    z    ∈    ℝ  , so, applying this definition to our question, we know that             |      2      x      +      3      |        ≥    0  , which is a stronger condition tan             |      2      x      +      3      |        ≥    -    13   (&quot;stronger&quot; means that             |      2      x      +      3      |        ≥    0   is more restrictive than             |      2      x      +      3      |        ≥    -    13  ). Instead of interpreting the issue as &quot;solve             |      2      x      +      3      |        ≥    -    13  &quot;, we are going to read it as &quot;solve             |      2      x      +      3      |        ≥    0  &quot; which, in fact, is easier to solve.In order to solve             |      2      x      +      3      |        ≥    0   we must again remember the definition of             |      z      |      , which is done by cases:If       z    ≥    0  , then             |      z      |        =    z  If       z    &amp;lt;    0  , then             |      z      |        =    -    z  Taking this into account for our problem, we can conclude that:If             (      2      x      +      3      )        ≥    0    ⇒          |      2      x      +      3      |        =    2    x    +    3   and then,             |      2      x      +      3      |        ≥    0    ⇒    2    x    +    3    ≥    0    ⇒    2    x    ≥    -    3    ⇒    x    ≥    -          3      2      If             (      2      x      +      3      )        &amp;lt;    0    ⇒          |      2      x      +      3      |        =    -          (      2      x      +      3      )       and then,             |      2      x      +      3      |        ≥    0    ⇒    -          (      2      x      +      3      )        ≥    0    ⇒    -    2    x    -    3    ≥    0    ⇒    -    2    x    ≥    3    ⇒    2    x    ≤    -    3   (observe that the sign of the inequality has changed on changing the sign of both members)       ⇒    x    ≤    -          3      2      As the result obtained in the first case is       ∀    x    ≥    -          3      2       and the result obtained in the second case is       ∀    x    ≤    -          3      2      , both put together give us the final result that the inequation is satisfied       ∀    x    ∈    ℝ  .

How to solve |2x+3|ge -13?

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