How to solve inequality with fractions?

Question

(x-3)/(x+2) < or equal to (x+5)/(x+4)

Answer 1

The inequality is

• Step-1

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

The exclude value of the inequality is - 2, - 4.

• Step - 2

Solve the related equation

Solution of related equation

• Step - 3

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

Answer 2

Continuous...

• 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

(-∞, -4)              x =  -5                         False

(-4, -11/3)          x = -3.8                   True

(-11/3, -2)          x = -3                               False

(-2, ∞)             x = -1                        True

Note that the original inequality contains a “≤ ” symbol, We inlude it into set of solutions at x = -11/3

Above statement is true.

x ≤ -11/3 is a solution of inequality.

The above conclude that the inequality is satisfied for all x - values in (-4, -11/3) and (-2, ∞)

This implies that the solution  of  the  inequality is  the  interval (-4, -11/3) and (-2, ∞). as shown in Figure below. Note that the original inequality contains a “≤ ” symbol. This means that the solution set contain the endpoint of the test interval is (-4, -11/3].

Solution of the inequality

-4 < x  ≤ -11/3

x  > - 2.