Topic 2 - 2.4  --  Linear Inequalities

Solve for the variable:

  a  -  3  >  9                     Add three to both sides if the inequality.

         3      3

    a       >   12

{a | a > 12}                     Answer in set builder notation. This says all a such that (|)

                                      a is greater than 12.

(12,  ∞)                           Answer in interval notation. (used in higher Math.) This

                                      says any number from 12, not including 12 to infinity.( ∞ )

Graph          The graph on the number line has a open circle on 12 and an arrow

                    pointing to the right.

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  -2m  <  24                     Divide both sides of the inequality by -2 and remember

   -2        -2                      when multiplying or dividing both sides of an inequality

   m   >   -12                    by a negative number, the inequality changes direction.

 {m | m > -12}               (turned around).

   [-12,  ∞)                       The bracket means the -12 is included in the answer.

                                        It comes from the equal in the problem.

Graph:         The graph on the number line has a closed circle on -12 and an

                    arrow pointing to the right.                

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  6  -  3y  >  9                    Add a -6 to both sides of the inequality.

 -6              -6

       - 3y  >  3                    Divide by -3. (turn the inequality around)

        -3       -3

          y   <  -1

    {y | y  <  -1}

    (-∞ ,  -1]

  Graph:        The graph on the number line has a closed circle on

                    -1 and an arrow pointing to the left.                           

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    -6  <  y  +  5  <  13        This is a compound inequality.  Add -5 to the three parts.

    -5             -5       -5

    -11 <      y     <    8  

  {y | -11 < y  <  8}

         [-11, 8]

Graph:          The graph has closed circles on -11 and 8 and shading          

                     between them.

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    -15  <  8  -  4k  <  -8        Add -8 to the three parts of the inequality.

     -8     -8               -8        

     -23  <    -4k    <   -16      Divide all three parts by -4 and turn both inequalities

      -4          -4           -4       around.

    23/4    >   k    >   4         Turn the whole inequality around. It is easier to read

         4   <    k    <   23/4     and graph.

   {k | 4 < k <  23/4}

      [4, 23/4]

Graph:     The graph on the number line has solid circles on 4 and 23/4 and the

                between them is shaded