\

1) The rational compound inequality is - 2 < - 8/x < 4.

\

Write the rational compound inequality using the word and. Then solve each inequality.

\

- 2 < - 8/x and  - 8/x < 4.

\

\
• Step-1
\

\

State the exclude values,These are the values for which denominator is zero.

\

The exclude value of the inequality is 0.

\

\
• Step - 2
\

\

Solve the related equation.

\

- 2 = - 8/x and  - 8/x = 4.

\

- 2x = - 8 and  - 8 = 4x.

\

x = 4 and  x = - 2.

\

Solutions of related equation x = 4 and  x = - 2.

\

\
• Step-3
\

\

Draw the vertical lines at the exclude value(x = 0) and at the solution (x = 4 and  x = - 2) to separate the number line into intervals.

\

\

\
• Step-4
\

\

Now test a sample value in each interval to determine whether values in the interval satisify the original inequality - 2 < - 8/x < 4.

\

Test Iinterval            x - value                              Inequality                                Conclusion

\

         x = - 3           - 2 < - 8/(- 3) < 4 ⇒ - 2 < 2.67 < 4                True.

\

(- 2, 0)                      x = - 1           - 2 < - 8/(- 1) < 4 ⇒ - 2 < 8 < 4                      False.

\

(0, 4)                        x = 1             - 2 < - 8/(1) < 4   ⇒ - 2 < - 8 < 4                      False.

\

(4, ∞)                        x = 5              - 2 < - 8/(5) < 4   ⇒ - 2 < 1.6 < 4                    True.

\

The above conclude that the inequality is satisfied for all x - values in (- ∞, - 2) and (4, ∞). This  implies  that  the  solution  of  the  inequality - 2 < - 8/x < 4 is  the  interval (- ∞, - 2) U (4, ∞), as shown in Figure 1.31. Note that the original inequality contains a “less than” symbol. This means that the solution set does not contain the endpoints of the test intervals are (- ∞, - 2) and (4, ∞).

\

The statement is true for x  = -0.1 and x  = 0.05.Therefore the solution is  x  < -1/15 and x  >0.

\

\

Solution x  < -1/15

\

x  > 0.

\