Solving Inequalities

Question

Please show the steps to solve the following inequalities.

 

1) (x² + x - 12)/x ≥ 0

 

2) 1/x > -1/4 

 

3) (x + 12)/(x + 7) < 7 

 

Also what is the difference between {and} and {or} when referring to the set of points that satisfy the inequality? An example would help a lot. 

 

Thanks in advance. 

Answer 1

1. x cannot be 0 (see denominator)

2. factor the numerator: (x+4)(x-3)

3. on (-infinity,-4) ALL three parts (two factors in numerator, and the denominator) are negative, and so this rational algebraic expression is negative.

4. on (4,0) one factor is positive, two others are negative, so overall expression is positive, and it's also zero when x=-4.

5. on (0,3) one factor is negative, others are positive, so overall expression is negative; also it's zero if x=3.

6. on (3,infinity) all factors are positive so overall expression is positive.

 

Assembling these facts, we get the solution set [-4,0) ∪ [3,infinity)

 

Of course ∪ (union) means OR.

 

The symbol ∩ (and, or intersection) is an operator between two sets such that the resulting set contains ONLY THOSE ELEMENTS FOUND IN BOTH OPERANDS (two sets being intersected).

 

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