solving x with inequalities "x" as denominator

Question

how to solve x with double inequalities "X" as a denominator.

for example:    -2<-8/x<4

Answer 1

Given inequlity -2<-8/x<4

-2< -8/x and -8/x < 4

-2< -8/x

Multiple to each side by x.

-2x < -8x/x

-2x < -8

Divide to each side by negitive 2 and change inequality symbol direction.

-2x/-2 > -8/-2

x > 4

Now -8/x < 4

Multiple to each side by x.

-8x/x < 4x

-8 < 4x

Divide to each side by 4.

-8/4 < 4x/4

-2 < x

Solution is x > -2 and x > 4.

Solution set is {x/x>-2}

 

Comments

The solution of the inequality - 2 < - 8/x < 4 is x < - 2 or x > 4.

The interval notation from of solution is (- ∞, - 2) U (4, ∞)

Answer 2

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

(- ∞, - 2)                   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 below. 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 solution of the inequality - 2 < - 8/x < 4 is x < - 2 or x > 4.

The interval notation from of solution is (- ∞, - 2) U (4, ∞).