# Solve Rational Inequalities - More Examples

More examples on solving rational inequalities.

__ Example 4__ Solve the rational inequality given by

**Solution to Example 4**

Given
Subtract (x - 4) / (x - 3) from both sides of the given inequality so that its right side equal to zero.

Rewrite the two rational expressions on the left side with common denominator.

Add the two rational expressions and simplify the numerator to obtain

Zeros of numerator and denominator

The numerator is a quadratic expression of the form ax^{2} + bx + c whose discriminant , b^{2} - 4(a)(c) = (-3)^{2} - 4(1)(5) = -11 , is negative and therefore has no zeros.

The zeros of the denominator: (x + 2)(x - 3) = 0 are x = -2 and x = 3.

The two zeros divide the real number line into 3 intervals as follows:

Select test values that are within each of the 3 intervals above and test the rational expression

a) interval (- ∞ , -2)

test value x = - 3

We now evaluate

b) interval (-2 , 3)

test value x = 0

We now evaluate

c) interval (3 , ∞)

test value x = 4

We now evaluate

We now put all the above in a table of sign.

__ Conclusion__ The solution set of the given rational inequality is given by the interval

__Graphical solution to the inequality.__

In the first step above, we obtained the inequality

*y*is positive over the interval

__ Example 5__ Solve the rational inequality given by

**Solution to Example 5**

Subtract 1 / (x + 3) from both sides of the inequality so that its right side equal to zero.

Factor the quadratic expression x^{2} - 2 x - 3 in the denominator.

Rewrite the rational the two expressions on the left side with the common denominator (x - 3)(x + 1)(x + 3) .

Add and simplify the two rational expressions on the left side of the inequality and simplify the numerator to obtain

Zeros of numerator and denominator

The zero of the numerator: 5x + 3 = 0, x = -3/5.

The zeros of the denominator: (x - 3)(x + 1)(x + 3) = 0 are x = 3, x = -1 and x = -3.

The four zeros divide the real number line into 5 intervals as follows:

Select and test values that are within each interval and test the rational expression

a) interval (- ∞ , -3)

test value x = -4

We now evaluate

b) interval (-3 , -1)

test value x = -2

We now evaluate

c) interval (-1 , -3/5)

test value x = -0.8

We now evaluate

d) interval (-3/5 , 3)

test value x = 0

We now evaluate

d) interval (3 , + ∞)

test value x = 4

We now evaluate

Let us now put all the above results in a table of sign

__ Conclusion__ The solution set of the given rational inequality is given by the interval

__Graphical solution to the inequality.__

We obtained above an equivalent (to the given) inequality to solve , which is

Below is shown the graph to the function

*y*is negative or zero over the interval

__ Example 6__ Solve the rational inequality given by

**Solution to Example 6**

Write the inequality with the
right side equal to zero by subtracting

1 / 3 from both sides.

Rewrite the left side of the inequality so that all terms have common denominator.

Group

The zeros of the numerator are found by solving two equations: (the expression within the absolute value symbol may be positive or negative)

1) 9x^{2} - (x^{2} - 3x - 4) = 0

8x^{2}+ 3x + 4 = 0 , discriminant = 3^{2} - 4(8)(4) < 0 , this equation has no real solutions

2) 9x^{2} - (-1)(x^{2} - 3x - 4) = 0

10x^{2} - 3x - 4 = 0 , two solutions: x = -1/2 and x = 4/5.

The zeros of the denominator: x^{2} - 3x - 4 = 0 are x = - 1 and x = 4

The four zeros divide the real number line into five intervals (Note that the zeros are ordered from the smallest to the largest).

Select test values that are within each interval and use them to find the sign of the expression

a) interval (- ∞ , -1)

test value x = - 2

We now evaluate

b) interval (-1 , -1/2)

test value x = - 0.75

We now evaluate

c) interval (-1/2 , 4/5)

test value x = 0

We now evaluate

d) interval (4/5 , 4)

test value x = 1

We now evaluate

d) interval (4 , + ∞)

test value x = 5

We now evaluate

Let us now put all the above results in a table of sign.

__The solution set of the given rational inequality is given by the interval__

**Conclusion**
__Graphical solution to the inequality.__

Shown below is an equivalent (to the given) inequality to solve which was obtained in step 1.

Below is shown the graph to the function

*y*is positive or zero over the interval