x = 12 is the smallest positive integer that satisfies the inequality
Step-by-step explanation:
The inequality that we have to solve is:
(5+x)(11-x)<0
First of all, we look at the solutions of the inequality
(5+x)(11-x)=0
We see that this equation has two roots:
x = -5
x = 11
Then we analyze separately each term of the product:
(5+x): this term is positive for x > -5 and negative for x < -5
(11-x): this term is positive for x < 11 and negative for x > 11
The global equation must be negative: in order to have a negative result, the product of the two brackets must be negative - therefore, one term must be positive and the other term must be negative, or viceversa.
This means that the inequality is satisfied in the following cases:
1) x > -5 and x > 11, which means simply x > 11
2) x < -5 and x < 11, which means simply x < - 5
Therefore, the smallest integer x that satisfies the inequality is x = 12 (if we are talking about positive numbers only; for negative number, there is no smallest number possible, because all numbers with x < -5 satisfies the inequality.
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