Solve the rational inequality $\frac{a+3}{a}<0$. Express the answer in interval form.
A. $[-3,0)$
B. $(0,3)$
C. $(-3,0)$
D. $[0,3)$
Answer (2)
The solution to the rational inequality a a + 3 < 0 is the interval ( − 3 , 0 ) . This was determined by finding the critical points, testing intervals, and identifying where the inequality holds. So, the correct multiple choice answer is C. ( − 3 , 0 ) .
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Find the critical points of the rational inequality by setting the numerator and denominator to zero: a + 3 = 0 gives a = − 3 , and a = 0 .
Test the intervals ( − ∞ , − 3 ) , ( − 3 , 0 ) , and ( 0 , ∞ ) using test points to determine where the inequality a a + 3 < 0 holds.
The inequality holds only in the interval ( − 3 , 0 ) .
Since the inequality is strict, exclude the endpoints. The solution is ( − 3 , 0 ) .
Explanation
Understanding the Problem We are given the rational inequality a a + 3 < 0 . Our goal is to find all values of a that satisfy this inequality and express the solution in interval notation.
Finding Critical Points and Intervals To solve this rational inequality, we first need to find the critical points. These are the values of a that make the numerator or the denominator equal to zero. The numerator is a + 3 , so we set a + 3 = 0 to find a = − 3 . The denominator is a , so we set a = 0 to find a = 0 . These critical points divide the number line into three intervals: ( − ∞ , − 3 ) , ( − 3 , 0 ) , and ( 0 , ∞ ) .
Testing the Intervals Now we need to test each interval to see where the inequality a a + 3 < 0 holds. We pick a test value in each interval and plug it into the inequality.
Interval ( − ∞ , − 3 ) : Let's choose a = − 4 . Then a a + 3 = − 4 − 4 + 3 = − 4 − 1 = 4 1 . Since 0"> 4 1 > 0 , the inequality does not hold in this interval.
Interval ( − 3 , 0 ) : Let's choose a = − 1 . Then a a + 3 = − 1 − 1 + 3 = − 1 2 = − 2 . Since − 2 < 0 , the inequality holds in this interval.
Interval ( 0 , ∞ ) : Let's choose a = 1 . Then a a + 3 = 1 1 + 3 = 1 4 = 4 . Since 0"> 4 > 0 , the inequality does not hold in this interval.
Determining the Solution The inequality a a + 3 < 0 holds only in the interval ( − 3 , 0 ) . Since the inequality is strict (i.e., < and not ≤ ), we do not include the endpoints a = − 3 and a = 0 in the solution. Therefore, the solution to the inequality is the interval ( − 3 , 0 ) .
Final Answer The solution to the inequality a a + 3 < 0 is the interval ( − 3 , 0 ) .
Examples
Rational inequalities are useful in various real-world scenarios. For example, they can be used to model situations where a quantity must be kept within a certain range, such as in chemical reactions where the concentration of a substance must be below a certain threshold to prevent unwanted side effects. They also appear in economics when analyzing cost-benefit ratios or in physics when dealing with rates and proportions. Understanding how to solve these inequalities helps in making informed decisions in these contexts.
