Solving inequalities?

Question

Can someone show me how to solve this?

Answer 1

The inequality is

Step-1 Finding key points:

Find zeros and undefined values of rational expression.

Equate the numerator to zero.

Equate the denominator to zero.

Subtract 3 from each side.

Step - 2

Solve the related equation.

Solution of related equation x = 4.

Step - 3

Draw the vertical lines at the exclude value and at the solution to separate the number line into intervals.

 

Answer 2

Step - 4

Now test  sample values in each interval to determine whether values in the interval satisify the inequality.

Test interval     x - value      Inequality                                          Conclusion

(-∞, -3)              x =  -4          (4-(-4)) / ((-4)^2+6(-4)+9)≥0

                                             8 / 1 ≥ 0 ⇒ 8 ≥ 0                                        True

(-3, 4)                x = -1           (4-(-1)) / ((-1)^2+6(-1)+9)≥0

                                             5 / 4 ≥ 0 ⇒ 1.25 ≥ 0                                   True

(4, ∞)                x = 5             (4-(5)) / ((5)^2+6(5)+9)≥0

                                             -1 / 64 ≥ 0 ⇒0.0156 ≥ 0                             False

The above conclude that the inequality is satisfied for all x - values in (-∞, -3) and (-3, 4].

This implies that the solution  of  the  inequality is  the  interval (-∞, -3) and (-3, 4]

as shown in Figure below. Note that the original inequality contains a “greater than or equal to” symbol. This means that the solution set contain the endpoints of the test intervals are (-∞, -3) and (-3, 4].

The solution of the inequality is x  ≤ 4 and x ≠ -3.

The interval notation from of solution is (-∞, -3) U (-3, 4].