Examples With Solutions

Rational Inequalities are solved in the examples below. Knowing that the sign of an algebraic expression changes at its zeros of odd multiplicity, solving an inequality may be reduced to finding the sign of an algebraic expression within intervals defined by the zeros of the expression in question.

-∞ | - 1 | 2 | +∞ |

Select a value of x in any of the intervals and use it to find the sign of the rational expression. Example for x = -3 in the interval (-∞ , -1), the rational expression (x - 2)/(x + 1) = (- 3 - 2)/(- 3 + 1) = 5 / 2. Hence the rational expression (x - 2)/(x + 1) is positive on the interval (-∞ , -1) .

-∞ | + | - 1 | 2 | +∞ |

-∞ | + | - 1 | - | 2 | + | +∞ |

Use x + 3 as a common denominator to rewrite the left side of the inequality as a single rational expressions as follows.

add the two rational expressions to obtain

We arrange the zeros of the numerator and the denominator on the number line from smallest to the largest as follows.

-∞ | - 5 | - 3 | +∞ |

Select a value of x in the interval (-∞ , - 5) and use it to find the sign of the rational expression. Example for x = - 6, the rational expression (-x - 5)(x + 3) = (6 - 5)/(- 6 + 3) = -1 / 3. Hence the rational expression (-x - 5)(x + 3) is negative on the interval (-∞ , - 5) .

-∞ | - | - 5 | - 3 | +∞ |

The zeros - 5 and - 3 are of odd multiplicity and therefore the sign of (-x - 5)(x + 3) will change at both zeros. Hence the signs of the expression (-x - 5)(x + 3) as we go from left to right are

-∞ | - | - 5 | + | - 3 | - | +∞ |

The solution set of the inequality is given by the union of all intervals where (-x - 5)(x + 3) is negative or equal to 0. Hence the solution set for the above inequality, in interval notation, is given by:

4 x

Rewrite the given inequality as

Arrange the zeros of the numerator and the denominator on the number and in order from smallest to the largest as follows.

-∞ | - 9 / 4 | - 2 | 1 | 3 | +∞ |

Select a value of x in the interval (-∞ , - 9/4) and use it to find the sign of the rational expression. Example for x = - 3, the rational expression ( (4x + 9)(x - 1) ) / ( (x - 3)(x + 2) ) = 2. Hence the rational expression ( (4x + 9)(x - 1)) / ( (x - 3)(x + 2) ) is positive on the interval (-∞ , - 9/4) .

All the zeros are of odd multiplicity and therefore the sign of ( (4x + 9)(x - 1) ) / ( (x - 3)(x + 2) ) will change at all zeros. Hence the signs of the expression ( (4x + 9)(x - 1) ) / ( (x - 3)(x + 2) ) as we go from left to right are

-∞ | + | - 9 / 4 | - | - 2 | + | 1 | - | 3 | + | +∞ |

The solution set of the inequality is given by the union of all intervals where ( (4x + 9)(x - 1) ) / ( (x - 3)(x + 2) ) is positive or equal to 0. Hence the solution set for the above inequality, in interval notation, is given by:

-∞ | - 6 | - 4 | 2 | +∞ |

The sign of ( (x+4)

-∞ | + | - 6 | - | - 4 | - | 2 | + | +∞ |

The solution set of the inequality is given by the union of all intervals where ( (x+4)

Solution:

Factor the numerator and the denominator of the given inequality.

x

Rewrite the given inequality as follows

The rational expression on the left side of the inequality has the zeros: - 3, - 1 and 3 ( x

-∞ | - 3 | - 1 | 3 | +∞ |

Select a value of x in the interval (-∞ , - 3) and use it to find the sign of the rational expression. Example for x = - 4, the rational expression ( x

The sign of the rational expression on the left side of the given inequality will change at the zeros - 1 and 3 because the are of odd multiplicity. Hence the signs of the expression rational expression as we go from left to right are

-∞ | - | - 3 | + | - 1 | - | 3 | + | +∞ |

The solution set of the inequality is given by the union of all intervals where rational expression on the left side of the given inequality is negative. Hence the solution set for the above inequality, in interval notation, is given by:

Find the LCD (least common denominator) of (x - 2)(x + 3) and x - 2 which is (x - 2)(x + 3).

Rewrite with common LCD.

Add the rational expressions on the left and simplify.

The rational expression on the left side of the inequality has the zeros at : - 3, 1 - √14 , 2 and 1 + √14. Arrange all the zeros on the number and in order from smallest to the largest as follows.

-∞ | - 3 | | 2 | +∞ |

Select a value of x in the interval (-∞ , - 3) and use it to find the sign of the rational expression. Example for x = - 4, the rational expression (-x

The sign of the rational expression on the left side of the given inequality will change at all the zeros because they all have odd multiplicity. Hence the signs of the expression rational expression as we go from left to right are

-∞ | - | - 3 | + | | - | 2 | + | - | +∞ |

The solution set of the inequality is given by:

ln(x + 1) is defined for x > - 1 hence any solution set must satisfy the condition x > - 1. (x + 2)/(x - 1) has a vertical asymptote at x = 1. From the two graphs, (x + 2)/(x - 1) (green) is greater than or equal to ln(x + 1) (red) for

(- 1, -0.59] ∪ (1 , 4.88]

Solve Rational Inequalities - More Examples

Rational Expressions

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Middle School Maths (Grades 6, 7, 8, 9) - Free Questions and Problems With Answers

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